1

Lecture Notes in Mathematics

Edited by A. Dold and B. Eckmann

1148

-------------------------------------------------------------

Mark A. Kon

Probability Distributions in Quantum Statistical Mechanics------------------------------------------------------------- Springer-Verlag Berlin Heidelberg New York Tokyo

2

Author

Mark A. KonDepartment of Mathematics, Boston University111 Cummington StreetBoston, MA 02215, USA

Mathematics Subject Classification: 82A 15, 60E07, 60G60, 46L60

ISBN 3-540-15690-9 Springer-Verlag Berlin Heidelberg New York TokyoISBN 0-387-15690-9 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned , specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproductionby photocopying machine or similar means, and storage in data banks. Under §54 of the German CopyrightLaws where copies are made for other than private use, a fee is payable to "VerwertungsgesellschaftWort", Munich.

© Springer-Verlag Berlin-HeidelbergPrinted in Germany

Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.2146/3140-543210

1

Preface

The purpose of this work is twofold: to provide a rigorous mathematical foundationfor study of the probability distributions of observables in quantum statistical mechanics,and to apply the theory to examples of physical interest. Although the work willprimarily interest mathematicians and mathematical physicists, I believe that results ofpurely physical interest (and at least one rather surprising result) are here as well. Indeed,some (§9.5) have been applied (see [JKS]) to study a model of the effect of angularmomentum on the frequency distribution of the cosmic background radiation. It issomewhat incongruous that in the half century since the development of quantumstatistics, the questions of probability distributions in so probabilistic a theory have beenaddressed so seldom. Credit is due to the Russian mathematician Y.A. Khinchin, whoseMathematical Foundations of Quantum Statistics was the first comprehensive work (tomy knowledge) to address the subject.

Chapters 7 and 8 are a digression into probability theory whose physical applicationsappear in Chapter 9. These chapters may be read independently for their probabilisticcontent. I have tried wherever possible to make the functional analytic and operatortheoretic content independent of the probabilistic content, to make it accessible to a largergroup of mathematicians (and hopefully physicists).

My thanks go to I.E. Segal, whose ideas initiated this work and whose work hasprovided many of the results needed to draw up the framework developed here. Mythanks go also to Thomas Orowan, who saw the input and revision of this manuscript,using TEX, from beginning to end; his work was invariably fast and reliable. Finally Iwould like to express my appreciation to the Laboratory for Computer Science at M.I.T.,on whose computers this manuscript was compiled, revised, and edited.

2

Contents

Preface 1

Ch. 1 Introduction 4 1.1 Purposes and Background 4 1.2 The Free Boson and Fermion Fields Over a Hilbert Space 7 1.3 The State Probability Space 10

Ch. 2 Value Functions on a Canonical Ensemble 13 2.1 Physical Interpretation of Value Functions 13 2.2 Distribution of Occupation Numbers 14 2.3 Formalism of Asymptotic Spectral Densities 16

Ch. 3 Integrals of Independent Random Variables 20 3.1 General Theory of One Parameter Families 20 3.2 Background for Singular Integrals 26 3.3 Distributions Under Symmetric Statistics 27 3.4 Antisymmetric Distributions 32

Ch. 4 Singular Central Limit Theorems, I 34 4.1 The Case 34JÐIÑ œ +I ß 1ÐIÑ ´ "#

4.2 The Case 38Ð1J Ñ Ð!Ñ !ß 1J Ð!Ñ œ !w w w

Ch. 5 Singular Central Limit Theorems, II 47 5.1 Asymptotics Under Uniform Spectral Density 47 5.2 Asymptotics for Non-Vanishing Densities Near Zero 51

Ch. 6 Physical Applications 60 6.1 The Spectral Measure: An Example in Schrödinger Theory 60 6.2 The Spectral Approximation Theorem 61 6.3 Observable Distributions in Minkowski Space 62 6.4 Distributions in Einstein Space 69 6.5 Physical Discussion 74

Ch. 7 The Lebesgue Integral 76 7.1 Definitions and Probabilistic Background 77 7.2 Definition of the Lebesgue integral 83 7.3 Basic Properties 88

3

Ch. 8 Integrability Criteria and Some Applications 94 8.1 The Criterion 95 8.2 The Riemann Integral 106 8.3 The -Integral 110V‡

Ch. 9 Joint Distributions and Applications 116 9.1 Integrals of Jointly Distributed R.V.'s 116 9.2 Abelian -algebras 117[‡

9.3 Computations with Value Functions 119 9.4 Joint Asymptotic Distributions 121 9.5 Applications 122

Epilogue 126References 127Index 130

List of Symbols 13'

4

Chapter 1

Introduction

§1.1 Purposes and BackgroundThe most general mathematical description of an equilibrium quantum system is in its

density operator, which contains all information relevant to the probability distributions ofassociated observables. Let be such a system, with Hamiltonian calculated in af Lreference frame with respect to which has zero mean linear and angular momentum.fLet be the inverse temperature of ( is temperature and is Boltzmann's" fœ X 5"

5X

constant). If the operator is trace class the appropriate density operator for is/ L" f

3 œ Þ/

/

L

L

"

"tr

Our general purpose is to obtain probability distributions of observables in fromfspectral properties of . If has very dense point spectrum, distributions depend on itL Lonly through an appropriate spectral measure; if the spectrum has a continuouscomponent then has infinite trace, so that the associated density operator and/ L"

probability distributions must again be defined via spectral measures. We will study howdistributions are determined by spectral measures and apply the resulting theory to the.(continuous spectrum) invariant relativistic Hamiltonian in Minkowski space and to the(discrete spectrum) invariant Hamiltonian in a spherical geometry (Einstein space), toderive probability distributions of certain important observables.

In some systems with continuous spectrum, a natural obtains through a net . ÖL ×% %!

of operators with pure point spectrum which in an appropriate sense approximates . InLorder that subsequent conclusions be well-founded, it must be required that be.independent of the choice of within some class of physically appropriate orÖL × ß% %!

“natural” nets. For example, if is an elliptic operator on a non-compact RiemannianEmanifold , a natural class arises in approximating by large compact manifolds. TheQ Qwell-developed theory of spectral asymptotics of pseudodifferential operators is usefulhere (see, e.g., [H], [See]). The procedure of infinite volume limits has also been studiedand developed in Schrödinger theory (see [Si, Section C6]). In physical applications suchas to the Planck law for photons, this approximation procedure is appropriate sincedistributions must be localized spatially, as well as with respect to wave propagationvector.

In the systems we consider, many interesting observables are sums of independentones indexed by the spectrum of a (maximal) commuting set of observables. Thus incases of continuous and asymptotically continuous spectrum, the notion of sum ofindependent random variables is very naturally replaced by that of an integral (over the

5

spectrum of observables), defined in a way completely analogous to the Riemann integral.The theory of such integrals and associated central limit theorems will be developed(Chapters 3-5), and then applied to particular random variables of interest.

A more comprehensive and abstract theory will be studied in Chapters 7-9. Theintegration procedure will be part of a more complete Lebesgue integration theory forrandom variable-valued functions. Connections will be made with the theory of randomdistributions and purely random fields [V,R1].

The related work in this area has been done primarily on , with Lebesgue measure.V8

General information on random distributions is contained in [GV]. Multi-dimensionalwhite noise was introduced in [Che], motivated by study of so-called Lévy Brownianmotion in [Lé]; the topic was further developed in [Mc]. A comprehensive theory ofgeneralized random fields was introduced by Malchan [M], using the notion ofbiorthogonality of random distributions on .V8

The non-linearity of the Lebesgue integral in Chapter 7 is not essential, since it isequivalent to an ordinary (linear) Lebesgue integral of a distribution-valued randomvariable field, and equivalently an integral over a space of logarithms of characteristicfunctions. Thus the integral is a simple and fundamental object. The probabilistic contentis itself novel and (hopefully) interesting, and these chapters have been written largely asa semi-autonomous part of the monograph. Probability and statistical mechanics are re-joined in Chapter 9.

There are two physical implications of this work which warrant attention. The first isthat non-normally distributed observables (such as photon number) can arise in physicallyattainable situations, namely those in which spectral density is non-vanishing near zeroenergy; the latter occurs in systems which are approximately one-dimensional, forexample, in optic fibers or wave guides. The second is that within the class of models inwhich the density operator depends only on the Hamiltonian, the "blackbody spectrum'' ina large-scale equilibrium system of photons admits an energy density spectrum whichfollows a classical Planck law, whose specific form depends only on basic geometry.This work provides a rigorous basis for the study of blackbody radiation in general spatialgeometries, specifically the spherical geometry of Einstein space. This is a model of thephysical universe (see [Se2]), and is useful in mathematical physics, being a naturalspatially compact space-time which admits the action of the full conformal group.

Some of the results to be presented here are treated somewhat differently and in morespecific situations in an excellent foundational work by Khinchin [Kh], who treatsasymptotic distributions for photon systems in Minkowski space. The asymptoticspectrum of the Hamiltonian there is approximated to be concentrated on the integers;such a procedure suffices for consideration of energy observables for photons in threedimensional geometries, although it must be considered somewhat heuristic. It ishowever not adequate for treatment of more general situations (as will be seen here), sincethe very volatile dependence of distributions on spectral densities near 0 is not visible insuch an analysis. In general terms, however, the present work extends many ideaspioneered by Khinchin.

The reading of any chapter is perhaps best done in two stages, the first involving onlybrief inspections of technical aspects of proofs, and the second involving a morethorough reading. Technicalities are often unavoidable in the proofs of theorems whosehypotheses are minimally technical.

6

We now provide a brief explanation of the structure of this monograph. Theremainder of this chapter provides a mathematical-physical framework. This includes adescription of Fock space, and a rigorous presentation of elementary results on quantumstatistical probability distributions. For example, we verify the common assumption thatoccupation numbers are independent random variables. Some of these results may befound (in somewhat less thorough form) in statistical physics texts. Chapter Two presentssome novel aspects of calculating distributions of observables (still in discrete situations),and presents fundamentals of the continuum limit. The basics of integration theory ofrandom variable-valued functions are developed in Chapter 3. Chapters 4 and 5 giveapplications to calculating observable distributions, including criteria for normality andnon-normality, and Chapter 6 provides physical applications. Here explicit distributionsare calculated, and a rigorous Planck law is derived for Bose and Fermi ensembles. It isshown that the Planck laws are essentially independent of the intrinsic geometry of largesystems. Chapters 7 and 8 develop an analogous Lebesgue theory of integration, andChapter 9 provides further applications, in the more general framework.

We dispose of a few technical preliminaries. Throughout this work (except in §9.5)we make the physical assumption that the chemical potential of particles underconsideration is 0; this does not involve essential loss of generality, as the general casecan be treated similarly. Probability distributions are studied for particles obeying Boseand Fermi statistics. The two situations are similar, and are studied in parallel. Theessentials of Segal's [Se3,4] formalism for free boson and fermion fields are usedthroughout and are described below.

We briefly mention some conventions. The symbols ™ ‚ ‘, , and denote theintegers and the natural, complex, and real numbers, respectively. The non-negativeintegers and reals are represented by and . The functions with compact™ ‘ _Gsupport are denoted by . The symbols , , and denote probability measure,G-

_ c X iexpectation and variance. The point mass at 0 is represented by , while and $! Ê Ò † Ódenote convergence in law (or weak convergence) and the greatest integer function,respectively. The spectrum of an operator is ; the words “positive” and “non-E ÐEÑ5negative” are used interchangeably, as are “increasing” and “non-decreasing”, etc.The abbreviations r.v., d.f., ch.f and a.s., mean “random variable,” “distributionfunction”, “characteristic function,” and “almost surely”. The notation w-lim denotes aweak limit, i.e., limit in distribution. denotes the normal distribution with mean RÐ+ß ,Ñ +and variance . If is an r.v. or a distribution, then denotes its d.f., \ J\

We use the notation if is bounded as . We use0ÐBÑ œ SÐBÑ ÐB Ä +Ñ B Ä +0ÐBÑSÐBÑ

0ÐBÑ œ 9ÐBÑ ÐB Ä +Ñ Ä ! B Ä + if as .9ÐBÑ0ÐBÑ

A question may arise as to the dimensions of physical quantities. The system of unitswill be entirely general, unless otherwise specified. For instance, the energy andIinverse temperature may be interpreted in any units inverse to each other. Also, a"preference for setting will be evident in various places.- œ "

7

§1.2 The Free Boson and Fermion Fields Over a Hilbert SpaceWe now present relevant aspects of free boson and fermion fields in their particlerepresentations. No proofs will be supplied; a more detailed and general description isgiven by Segal [Se3, Se4].

Let be a separable complex Hilbert space; for notational convenience and without[loss of generality assume is infinite dimensional. For let[ 8 −

^ [ [ [ [8

3œ"

8

œ Œ Œá Œ œ Ð"Þ"Ñ8be the -fold tensor product of with itself, and be the unitary representation of8 Z Ð † Ñ[ 8

ÐFÑ

the symmetric group of order on which is uniquely determined by the propertyD ^8 88

Z Ð Ñ B œ B Ð − à B − Ñà8ÐFÑ

3œ" 3œ"

8 8

3 8 3Ð3Ñ5 5 D [ 8 8 5"

Z Ð Ñ8 8ÐFÑ ÐFÑ

5 ^ thus permutes tensors. The (closed) subspace consisting of elements leftinvariant by is the ˜ ™Z Ð Ñ À − 88

ÐFÑ85 5 D -fold symmetrized tensor product of with[

itself.By convention ^ ‚!

ÐFÑœ Þ

For B ßá ß B −" 8 [

B ” B ”á ” B ´ B ´ Z Ð Ñ B Ð"Þ#Ñ"

8x" # 8 3 3

3œ" 3œ"

8 8

−8ÐFÑ2 " 8

5 D8

5

is the orthogonal projection of into ; the latter is clearly spanned by vectorsŒ3œ"

8

B3 8ÐFÑ

^

of the form (1.2). Note that for and B − − ß3 8[ 5 D

2 23œ" 3œ"

8 8

3 Ð3ÑB œ B Þ Ð5 1.3)

The direct sum of symmetrized tensor products over all orders

^ ^F

8œ!

_

8ÐFÑ´ :

is the over .(Hilbert) space of symmetrized tensors [

Any unitary on can be lifted to a unitary which isY ÐYÑ À Ä[ > ^ ^8 8 8ÐFÑ ÐFÑ ÐFÑ

defined uniquely by

> [8ÐFÑ

3œ" 3œ"

8 8

3 3 3ÐY Ñ B œ ÐYB Ñ ÐB − Ñà Ð"Þ%Ñ2 2

8

> ^8 8ÐFÑ ÐFÑ

ÐY Ñ 8 Y is simply the restriction to of the -fold tensor product of with itself; weappend the convention that is the identity. We define> ‚ ‚!

ÐWÑÐY Ñ À Ä

> >F

8œ!

_

8ÐFÑÐY Ñ œ ÐYÑß:

so that for B − ß8 8^

> >F 8 8

8œ! 8œ!

_ _

8ÐFÑÐY Ñ B œ ÐYÑB Þ : :

A self-adjoint operator in is naturally lifted to as the self-adjoint generator of theE [ ^F

one parameter unitary group :>F3>EÐ/ Ñ

. ÐEÑ ´ Ð/ Ñ ß Ð"Þ&Ñ" .

> .>> >F F

Ð3>EÑ

>œ!¹

with the derivative taken in the strong operator topology.The definitions for antisymmetric statistics are fully analogous to those above. Let

=Ð Ñ − Z Ð † Ñ5 5 ^ denote the sign of , and be the unitary representation of on D D8 8 88ÐJÑ

defined by

Z Ð Ñ B œ =Ð Ñ B Ð − ß B − ÑÞ8ÐJÑ

3œ" 3œ"

8 8

3 8 3Ð3Ñ5 5 5 D [ 8 8 5"

Let denote the subspace of elements left invariant by , with^ 5 58 8ÐJÑ ÐJÑ

8˜ ™Z Ð Ñ À −D^ ‚!

ÐJÑ´ ; the collection

B • B •á • B ´ B ´ Z Ð Ñ B ÐB − Ñà Ð"Þ'Ñ"

8x" # 8 3 3 3

3œ" 3œ"

8 8

−8ÐJÑ4 " 8

5 D8

5 [

spans . The over is^ [8ÐJÑ space of antisymmetrized tensors

^ ^J

8œ!

_

8ÐJÑœ Þ Ð"Þ(Ñ:

If is unitary on , then is defined byY ÐYÑ À Ä[ > ^ ^8 8 8ÐJÑ ÐJÑ ÐJÑ

>8ÐJÑ

3œ" 3œ"

8 8

3 3ÐY Ñ B œ YB ß 4 4with the identity,> ‚ ‚!

ÐJÑÀ Ä

9

> >J

8œ!

_

8ÐJÑÐY Ñ ´ ÐYÑ:

is the lifting of to . If is self-adjoint in , then is the generator of theY E . ÐEÑ^ [ >8ÐJÑ

J

unitary group .>J3>EÐ/ Ñ

Henceforth statements not specifically referring to the (anti-)symmetric constructionswill hold under both statistics; in particular this will hold when subscripts and areF Jomitted.

Definition 1.1: The operator is the of .. ÐEÑ E> quantization

The map acts linearly on bounded and unbounded self-adjoint operators to the.>extent that if are mutually orthogonal projections in and , thenÖT × ÖI × §4 4− 4 4− [ ‘

. I T œ I . ÐT Ñà Ð"Þ)Ñ> > " "4œ" 4œ"

_ _

4 4 4 4

this fact will later prove useful. Physically, is the Hilbert space of statesÐ ß . ÐEÑÑ^ >together with the Hamiltonian of a many-particle non-interacting system each particle ofwhich has states in and time evolution governed by .[ E

If is an orthonormal basis for then orthonormal bases for are givenK œ Ö1 ×4 4− 8 [ ^by

U

U

8ÐFÑ

3œ"

8

4 " # 8 3

8ÐJÑ

3œ"

8

4 " # 8 3

œ 1 À 4 Ÿ 4 Ÿ á Ÿ 4 à 4 −

œ 1 À 4 4 á 4 à 4 −

Ÿ2 Ÿ4

3

3

(symmetric)

(antisymmetric) .

Ð"Þ*Ñ

The basis can be represented through the correspondenceU8ÐFÑ

23œ"

8

4 " # 41 Ç Ð7 ß7 ßáÑ œ Ð1 − KÑß3 3

n

where denotes the number of appearances of on the left hand side, and 7 1 7 œ 8Þ3 3 33œ"

_!If we append the convention , thenF œ Ö"× −!

ÐFÑ‚

a ^˜ œ F §.8œ!

_

8ÐFÑ

F

is an orthonormal basis for We can thus write^FÞ

10

a ™˜ œ Ð7 ß7 ßáÑ À 7 − à 7 _ ß Ð"Þ"!+Ñ Ÿ"" # 3 3

3œ"

_

where corresponds to 1, the basis for a ^ ‚œ Ð!ß !ßá Ñ œ Þ!ÐFÑ

Correspondingly, if

a ^s œ F § ß.8œ!

_

8ÐJÑ

J

there is a bijection

43œ"

8

4 " #1 Ç Ð7 ß7 ßáÑ œ3

n;

note that no above may be greater than 1, since 1 limits the number of appearances of73

13 under antisymmetric statistics (see (1.6)). The collection

as œ Ð7 ß7 ßáÑ À 7 − Ö!ß "×à 7 _ Ð"Þ"!,Ñ Ÿ"" # 3 3

3œ"

_

is a basis of .^J

§1.3 The State Probability SpaceLet be an orthonormal basis for , with the orthogonal projectionK œ Ö1 × T4 4− 4 [

onto the span of . If is the basis of constructed in (1.10), and14 a ^n œ Ð7 ß7 ßáÑ − ß" # a then

. ÐT Ñ œ 7 Ð4 − ÑÞ Ð"Þ""Ñ> 4 4n n

Consequently if and then by (1.8)Ö- × § G œ - T4 4− 4 44œ"

_

‘ !

. ÐGÑ œ - 7 Þ> n n "4œ"

_

4 4 (1.12)

Given a distinguished operator on such that is non-negative self-adjoint and3 ^ 3Ð3Ñ tr can be interpreted as a random variable on :Ð33Ñ œ "ß . ÐGÑ3 > a

Definition 1.2: An operator with properties and is a on .3 ^Ð3Ñ Ð33Ñ density operator Through the probability measure ( , on , it forms the c 3 an n nÑ œ Ø Ù state probabilityspace value corresponding to . The r.v. (if defined) is the Ð ß Ñ -Ð Ñ œ Ø. ÐGÑ ß Ùa c 3 >n n nfunction of or of .. ÐGÑ G>

If is self adjoint on , , and is trace class,E ! /[ " . ÐEÑ" >

11

3 œ/

/

. ÐEÑ

. ÐEÑ

" >

" >tr

is a density operator. If diagonalizes and is as above, then the stateK œ Ö1 × E4 4− aprobability space corresponding to is the or Ð ß Ña c 3 (symmetric antisymmetric)canonical ensemble over . Clearly any operator which commutes with defines anE G Er.v. in this way. These will be our object of study; of particular interest will be valuefunctions of number operators R œ . ÐT ÑÞ4 4>

In physical applications is a positive Hamiltonian governing time evolution ofEsingle particles in a multi-particle system. If , then represents aE1 œ I 1 1 − K4 4 4 4

physical single particle state of energy is the many-particle stateI à œ Ð7 ß7 áÑ4 " #ßnwith particles in state . By (1.11), the observable is the number of7 1 Ð3 œ "ß #ßá Ñ R3 3 4

particles in state , and its value function is interpreted accordingly.14We now derive necessary and sufficient conditions for existence of a canonical

ensemble over , which we assume to be positive in . The following proposition hasE [been implicit in the physical literature on statistical mechanics.

Definition 1.3: An operator is 0- 0 is outside its spectrum.free if

Proposition 1.4: The operator is trace class if and only if/ . ÐEÑ" >

is trace class and is 0-free under symmetric statistics.Ð3Ñ / E E"

is trace class under antisymmetric statistics.Ð33Ñ / E"

Note that if either or is trace class, then both and have pure/ / E . ÐEÑ E . ÐEÑ" " > >point spectrum and finite multiplicities, since . Thus previous assumptionsE œ . ÐEÑl> ^"

on have involved no loss of generality.EWe sketch an argument for Proposition 1.4. In the symmetric case, define the

function on by . By (1.12)˜ n n n4 4 4a Ð Ñ œ 7

tr / œ / Þ Ð"Þ"$Ñ . ÐEÑ

−

I Ð Ñ" >

a

"F 4œ"

_

4 4"n

n n

˜

!

On the other hand, when the product

$ $ " 4œ" 4œ" 3œ!

_ _ _ I " IÐ" / Ñ œ / ß Ð"Þ"%Ñ" "4 4

is multiplied out, it coincides with (1.13), so the latter converges if and only if

12

"4œ!

_ I/ _ß" 4

and no are 0. The proof in the antisymmetric case is similar.I4

We thus have

O ´ / œ

O œ Ð" / Ñ

O œ Ð" / Ñtr . (1.15)

(symmetric)

(antisymmetric) . ÐEÑ

F4œ"

_ I "

J4œ"

_ I

" >

"

"

ÚÝÝÝÛÝÝÝÜ

##

4

4

Corollary 1.4.1: If is trace class, then has pure point spectrum and finite/ E . ÐEÑ" >

multiplicities.

13

Chapter 2

Value Functions on a Canonical Ensemble

Throughout this chapter is a positive self-adjoint (energy) operator in , satisfyingE [Ð3Ñ Ð33Ñ E ÖI × or of Proposition 1.4. In particular has pure point spectrum , and an4 4œ"

_

orthonormal eigenbasis , with . Assume is separable and (forK œ Ö1 × E1 œ I 14 4− 4 4 4 [notational convenience) infinite-dimensional, and define the number operator R œ .4 >ÐT Ñ K4 as before. The basis of corresponding to is given in (1.10); its generica ^elements will be . Definen œ Ð7 ß7 ßáÑ" #

3 œ Þ Ð#Þ"Ñ/

/

. ÐEÑ

. ÐEÑ

" >

" >tr

If is the value function of , and that of , then+ . ÐEÑ 8 R> 4 4

2Ð+Ñ œ 2ÐI Ñ8 Þ Ð#Þ#Ñ"4œ"

_

4 4

More generally, if commutes with and has value function G E G1 œ - 1 ß . ÐGÑ4 4 44œ"

_

> !- 84 4.

Definition 2.1: The r.v. is the in the canonical ensemble over8 44>2 occupation number

E.

Note that in the ensemble, according to (1.12),

c 3Ð Ñ œ Ø Ù / ß / Þ Ð#Þ$Ñ" "

/ On n n n n , = =

tr . ÐEÑ

I 7 I 7

" >

" "¢ £! !4œ"

_

4 4 4 4

§2.1 Physical Interpretation of Value FunctionsThe canonical ensemble over at inverse temperature describes a non-self-E !"

interacting system of particles in equilibrium at temperature ( is Boltzmann'sX œ 5"5"

constant) with particles whose individual time evolution is governed by the HamiltonianE G. If the operator , representing a physical observable for single particles, commuteswith , is the observable representing "total amount" of . Precisely,E . ÐGÑ G>

14

. ÐGÑ œ 7 ØG1 ß 1 Ù> n n "4œ"

_

3 3 3 , (2.4)

ØG1 ß 1 Ù G 14 4 4 representing the value of in the state .The value function of is a random variable on the canonical ensemble,. ÐGÑ>

representing the probabilistic nature of . By its definition. ÐGÑ>

-Ð Ñ œ 7 ØG1 ß 1 ÙÞ Ð#Þ&Ñn "4œ"

_

3 4 4

In particular, the occupation number corresponding to is the r.v. on which8 1 Ð ß Ñ4 4 a con takes the value , i.e., the number of particles in state . The value of n 7 1 œ I 84 4 4 4 4?on is and represents the "total energy" of the particles in state .n I 7 ß 14 4 4

§2.2 Distribution of Occupation NumbersIf denote occupation numbers and if and8 œ Ð> ß > ßá Ñß œ Ð8 ß 8 ßá Ñ3 " # " #t n

t n† œ > 8 Ö8 ×! 3 3 4 4−, then according to (2.3) the joint characteristic function of is

F XÐ Ñ œ Ð/ œ / ß Ð#Þ'Ñ"

Ot 3 † 3 † †

−

t n t n E n

n) "

a

"

where , and denotes mathematical expectation. The right side factorsE œ ÐI ßI ßáÑ" # Xto give

FÐ Ñ œ"

O

Ð" / Ñ

" / Ñt $œ

4œ"

_ I 3> "

I 3>

"

"

4 4

4 4

(symmetric)( (antisymmetric)

. (2.7)

Thus, occupation numbers are independent r.v.'s with characteristic functions84

F4

I I 3> "

I " Ð I 3>ÑÐ>Ñ œ Ð#Þ)Ñ

Ð" / ÑÐ" / Ñ Þ

Ð" / Ñ Ð" / Ñœ " "

" "

4 4

4 4

By inversion of (2.8), is geometric in the first case, with84

cÐ8 œ 6Ñ œ Ð" / Ñ/ Ð6 œ !ß "ßá Ñß Ð#Þ*Ñ4 I I 6" "4 4

X iÐ8 Ñ œ ß Ð8 Ñ œ ß Ð#Þ"!Ñ" /

/ " Ð/ "Ñ4 4I I

I

#" "

"

4 4

4

where denotes variance. Under antisymmetric statistics is Bernoulli, withi 84

15

cÐ8 œ 6Ñ œ Ð#Þ""ÑÐ/ "Ñ 6 œ "

Ð" / Ñ à 6 œ !4

I "

I "œ "

"

4

4

;

X iÐ8 Ñ œ ß Ð8 Ñ œ Þ" /

/ " Ð/ "Ñ4 4I I

I

#" "

"

4 4

4

The above distributions (specifically the independence of the occupation numbers)explicitly verify facts which have been used in the physical literature for some time.

Rewriting of (2.3) shows that if

a6 " # 4

4œ"

_

œ œ Ð7 ß7 ßáÑ À l l ´ 7 œ 6 ß Ð#Þ"#Ñ Ÿ" n n

the particle number value function satisfies8 œ 8!4œ"

_

4

c ™Ð8 œ 6Ñ œ / œ / Ð6 − Ñ Ð#Þ"$Ñ" "

O O" "n

E n

−

†

"Ÿ4 ŸáŸ4

I

a

""

" " 6

7œ"

6

47 !

under symmetric statistics, with replaced by in the antisymmetric case. There isŸ a method of expressing (2.13) more simply for small values of which, however, becomes6more complicated as increases, and is not discussed here.6

The total energy

? œ † œ I 8E n " 4 4

is a discrete r.v. with

TÐ œ ,Ñ œ / œ ß" / .Ð,Ñ

O O? "

n−

, ,

a

""

,

where is the cardinality of : .Ð,Ñ œ Ö − † œ ,×Þa a, n n EWhen has uniformly spaced spectrum , (physically interesting in one-E I œ 44 %

dimensional situations), these specialize in the symmetric case to

16

cÐ8 œ 6Ñ œ /"

O

œ / /"

O

œ /" /

O " /

F "Ÿ4 áŸ4

4

F "Ÿ4 áŸ4

4

4  4

4

F "Ÿ4 ŸáŸ4

4 4

"" " "

" 6

7œ"

6

7

" 6"

7œ"

6"

7

6 6"

6

" 6"

7œ"

6"

7 6"

"%

"%"%

"% "%

!

!

!

F "Ÿ4 ŸáŸ4

4 # 4

#

F

6 4 " 6 4

4œ"

6 _

4œ6"

"%

"% %"%

"% "%

"% "% "% "%

œ /" /

O Ð" / ÑÐ" / Ñ

œ á œ / Ð" / Ñ œ / Ð" / ÑÞ"

O

Ð#Þ"%

"$ $

" 6#

7œ"

6#

7 6#!

+Ñ

Similarly in the antisymmetric case

cÐ8 œ 6Ñ œ / Ð" / ÑÞ Ð#Þ"%,Ñ"

OJ 4œ"

6 4

6Ð6"Ñ# $ "%

Note that the tail of the number r.v. in (2.14) is geometric under symmetric statistics, andresembles a discretized Gaussian in the antisymmetric case.

It is easily seen in this case that

c ? %Ð œ 6Ñ œ

ÚÛÜ

/ :O

/ ;O

66

F 6

6

J

"%

"%

(symmetric)

(antisymmetric)(2.15)

where is the number of ways of representing as a sum of (distinct) positive: Ð; Ñ 66 6

integers. The asymptotics of the combinatorial functions and are tabulated in books: ;6 6

of mathematical functions (see [AS]).

§2.3 Formalism of Asymptotic Spectral DensitiesIn many situations in which a self-adjoint operator has large or asymptotically infinite

spectral density, use of a spectral measure on the real line is necessary for interpretation ofsums indexed by the spectrum, or mediating more detailed spectral information. Aclassical example occurs in Sturm-Liouville systems on , whose “natural” spectralÒ!ß_Ñmeasures for eigenfunction expansions are defined in terms of limits of spectral densitiesof their restrictions to intervals (see [GS]). In the study of sums such as , a8 œ 8! 4

similar situation occurs if the spectrum of is very dense or continuous, e.g., if ÖI × E E4

is an elliptic differential operator on a “large” Riemannian manifold. See Simon'sreview article [Si, Sections C6, C7] for a perspective on this problem in Schrödingertheory. Situations such as these certainly predominate in macroscopic systems. In such

17

cases, the appropriate probabilistic context for study of probability distributions incanonical ensembles involves integrals (rather than sums) of independent randomvariables with respect to appropriate spectral measures.

We begin with preliminary notions and facts. Henceforth will denote the greatestÒ † Óinteger function.

Definition 2.2: A is a -finite measure onspectral measure. 5 ‘ œ Ò!ß_ÓàJÐIÑ œ Ò!ßIÓ E. . is the of . The operator with point spectrumspectral function %

consisting of the discontinuities of , each discontinuity having multiplicity” •JÐIÑ% I

” • ” •JÐIÑ JÐI Ñ% %

, is the - corresponding to .% .discrete operator

The operator has eigenvalue density essentially given by E Þ%.%

The correspondence between spectral measures and nets is bijective.. ÖE ×% %!

Indeed, the -net of functions converges uniformly to , determining .% % .” •JÐIÑ% J

We now investigate when defines a canonical ensemble, using criteria establishedE%

in Chapter 1. We require the following convention: the domain of integration in aStieltjes integral includes endpoints, and'

+

,1ÐBÑ.0ÐBÑ

( (+ +

, ,

Ä!

1ÐBÑ.0ÐBÑ ´ 1ÐBÑ.0ÐBÑÞ Ð#Þ"'Ñlim% %

We begin with a technical lemma.

Lemma 2.3. Let and be defined on be monotone non-increasing0Ð † Ñ 1Ð † Ñ Ò+ß ,Óß 1Ð † Ñand non-negative, be monotone and be Stieltjes-integrable with respect to0Ð † Ñ 1Ð † Ñ0Ð † Ñ Ò0Ð † ÑÓÞ and

Then

º º( (+ +

, ,

1ÐBÑ.0ÐBÑ 1ÐBÑ.Ò0ÐBÑÓ Ÿ 1Ð+ÑÞ Ð#Þ"(Ñ

This holds if “a” is replaced by “ ”, “ ” by “ ” or if is replaced by+ , , Ò+ß ,Ó

Ò+ß_Ñ.

Proof: Assume without loss that is non-decreasing, and that is continuous on0Ð † Ñ 0Ð † ÑÒ+ß ,Ó , œ _; some inessential complications occur if the latter fails. Assume (otherwise0 1 and can be appropriately extended), and let

+ Ÿ B B B áß Ð#Þ")Ñ" # $

where are the discontinuity points of . If is empty the assertion isÖB × Ò0Ð † ÑÓ ÖB ×5 5

trivial. Otherwise let

18

W œ Ò+ß B Óß W ÐB ß B Óß W œ ÐB ß B Óßá Þ" " # " # $ # $

Then

( ( (" "+ W +

_ _

5œ" 5œ"

51ÐBÑ.0ÐBÑ œ 1ÐBÑ.0ÐBÑà 1ÐBÑ.Ò0ÐBÑÓ œ 1ÐB Ñß Ð#Þ"*Ñ5

and

1ÐB Ñ   1ÐBÑ.0ÐBÑ   1ÐB Ñ Ð5 œ "ß #ßá ÑÞ Ð#Þ#!Ñ5 5"W

(5"

Hence

º º" "(5œ" 5œ#

5W

1ÐB Ñ 1ÐBÑ.0ÐBÑ Ÿ 1Ð+ÑÞ Ð#Þ#"Ñ5

The fact that

(W"

1ÐBÑ.0ÐBÑ Ÿ 1Ð+Ñ Ð#Þ##Ñ

and the first inequality in (2.20) imply (2.18), the remaining assertions follow similarly. è

Corollary 2.4: If instead of monotone is of bounded variation, then1

» »( (+ +

, ,

1ÐBÑ.0ÐBÑ 1ÐBÑ.Ò0ÐBÑÓ Ÿ #1Ð+ÑÞ

Theorem 2.5: Let be a spectral measure and the corresponding -discrete. %E%

operator. Let be a non-increasing function. Then is trace class if and only if0Ð † Ñ 0ÐE Ñ%

(!

_

0ÐIÑ. ÐIÑ _Þ Ð#Þ#$Ñ.

Proof: If is trace class, then0ÐE Ñ%

tr 0ÐE Ñ œ 0ÐIÑ. ßJÐIÑ

% ( ” •!

_

%while

( ( ! !

_ _

0ÐIÑ. ÐIÑ œ 0ÐIÑ. Þ Ð#Þ#%ÑJÐIÑ

. %%

If is bounded on , the last expression is finite. Otherwise, the lemma impliesJÐ † Ñ Ò!ß_Ñ

19

% % %% %

( (” • ” •! !

_ _

0ÐIÑ. Ÿ 0ÐIÑ. 0Ð! Ñ œ Ð 0ÐE ÑÑ 0Ð! Ñ _ÞJÐIÑ JÐIÑ

tr %

Conversely if (2.23) holds, then

tr 0ÐE Ñ œ 0ÐIÑ. Ÿ 0ÐIÑ. 0Ð! Ñ _Þè Ð#Þ#&ÑJÐIÑ JÐIÑ

% ( (” • ! !

_ _

% %

Corollary 2.6: The operator is trace class for some if and only if it is for0ÐE Ñ !% %all .% !

Corollary 2.7: The operator is trace class if and only if/ E" %

(!

_ I/ . ÐIÑ _Þ" .

Definition 2.8: A spectral measure is 0- if 0.. .free ÐÖ!×Ñ œ

Note that is 0-free if and only if is 0-free for all This fact together with. %E Þ%

Corollary 2.7 and Proposition 1.4 completely characterizes those spectral measures whichdefine canonical ensembles.

20

Chapter 3

Integrals of Independent Random Variables

In the limit of continuous spectrum for a Hamiltonian , sums of r.v.'s indexed by theEspectrum, such as value functions of quantizations of observables commuting with , areEmost profitably represented as integrals for both notational and conceptual reasons. Wenow develop a theory and properties of these to the extent that they are useful later.These results will be in a more natural probabilistic setting in Chapters 7 and 8, whereintegrals of r.v.-valued functions over a measure space will be studied.

§3.1 General Theory of One Parameter FamiliesFor , let be a one-parameter family of independent r.v.'s and a spectralI   ! \ÐIÑ .

measure on . We integrate with respect to as follows. For each let‘ . % \ !Y% %œ ÖI ×4 4−N% be positive numbers (not necessarily distinct) such that the cardinalityR Ð+ß ,Ñ ÖÒ+ß ,Ó×% % of satisfiesY

% Ò .R Ð+ß ,Ñ Ò+ß ,Ó Ð$Þ"+Ñ%%Ä!

for all ( may be ). To avoid pathologies, assume also the existence of! Ÿ + , , _numbers such that for any interval whose measure exceeds Qß5 ! M Qß

% .R ÐMÑ Ÿ 5 ÐMÑÞ Ð$Þ",Ñ%

Definition 3.1: -net.Ö ×Y .% %! is a spectral

Given a function ( on positive , we define the probability distribution9 % %Ñ

( "‘ %

%

\ÐIÑ Ð. Ñ ´ Ð Ñ \ÐI Ñ9 . 9 %w- limÄ!

4

4

with sums on the right defined only if order-independent. The limit is in the topology ofconvergence in law, with the integral defined only if independent of .Y%

If is real-valued on the Riemann-Stieltjes integral of with respect to can be0 ß 0‘ .

defined as

V

( "‘ .

0ÐIÑ. ´ 0ÐI Ñ Ð I Ñ Ð$Þ#Ñ. . ?limÐT ÑÄ!

3 3

with a partition of into intervals , , and theT I I − I ß ÐT Ñ œ Ð I Ñ‘ ? ? . . ?3 3 3 3 3sup

limit is defined only when independent of and , as well as order of summation (i.e.,T I3

convergence is absolute). In evaluating the right side of (3.2) and all similar sums we

21

apply the convention: if has an atom at , a partition may formally divide . I   ! T Iinto degenerate intervals , each consisting of just the point , such thatÖ I × I? 3 3 N%!N

3. ? .Ð I Ñ œ ÐÖI×Ñà the union of such an interval with an adjacent non-degenerate

interval is also allowable. is an if , except possibly whenT ÐI Ñ œ ÐI Ñeven partition . .3 4

I4 is a rightmost non-empty interval, which is allowed smaller measure.

Theorem 3.2: If is a real number (i.e., a point mass) for each , then \ÐIÑ I \ÐIÑ.' .

coincides exactly with .V' \ÐIÑ..

Proof: Assume that exists, and let be a spectral -net. ForV 4 4 N' \ÐIÑ. œ ÖI ×. Y .% % % %

8 − T œ Ö I × ? ‘ let be a sequence of even partitions of following the above8 83 3 M

% 8

conventions, such that . The indexing set may be finite or the whole. Ò ÐT Ñ ! M §8 88Ä_

of . For let satisfy˜ ?3 − M ß I ß I − I8 83 8383‡

\ÐI Ñ Ÿ \ÐIÑ Ÿ \ÐI Ñ ÐI − I Ñß Ð$Þ$Ñ" "

83 8383‡

# #83 83˜ ?

and define

+ œÐ I Ñ

838

3−M

83

#"

8

. ?

E œ \ÐI Ñ Ð I Ñß F œ \ÐI Ñ Ð I ÑÞ8 83 8 83 83

3 M 3 M83‡" "

% %8 8

. ? . ?˜

If has compact support then is finite, and by (3.1) and (3.3), the distance between the. M8number and the interval approaches 0 as 0. Since% %!

4−N4 8 8 8 8

%

\ÐI Ñ ÒE + ßF + Ó Ä%

+88Ä_Ò

ÒE + ßF + Ó \ÐIÑ. ß8 8 8 88Ä_ VÒ .(

and the result follows here, since the -net was arbitrary. For the general case, we. Y% must show

% Ò"lI l.

4

%4

l\ÐI Ñl !ß%. Ä _

uniformly in . Let be an even partition of such that% ? ‘T œ Ö I ×4

"4

4 4l\ÐI Ñl Ð I Ñ G Ð$Þ%Ñ. ?

for any set of By (3.1b)I − I Þ4 4?

22

% ? . ?R Ð I Ñ Ÿ 5 ÐQß Ð I ÑÑß% 4 4max

so that

% . ?"I − I

4 4I− I

%4 3 3?

%?

l\ÐI Ñl Ÿ 5 ÐQß Ð I ÑÑ l\ÐIÑlÞmax sup

Since is even and by (3.4),T

% . ? Ò" "lI l. I Ò.ß_ÓÁ

4 4I− I .Ä_

%4 4 4

l\ÐI Ñl Ÿ 5 l\ÐIÑl ÐQß Ð I ÑÑ !ß Ð$Þ&Ñ%

? 9 ?

sup max

convergence being manifestly uniform in .%Conversely, assume exists. Let be a sequence of partitions' \ÐIÑ. T œ Ö I ×. ?8 84

of with and let . There exists an even partition‘ . Ò ?8 84 84

8Ä_ÐT Ñ !ß I − I

T œ Ö I ׇ ‡4? such that

U œ l\ÐIÑl Ð I Ñ _à"4 I− I

4‡sup

? 4‡

. ?

hence if , then. .ÐT Ñ ÐT Ñ8‡

"4 I− I

83sup? 84

‡

l\ÐIÑl Ð I Ñ Ÿ $UÞ Ð$Þ'Ñ. ?

Let be chosen such that for each there is an integer (depending onlyY% % % %œ ÖI × I 83 3

on ) and a such that% 4

+Ñ Ö3 À I œ I × Ò † Ó The cardinality of is where denotes the greatest%. ?

%3 8 4Ð I Ñ

%%’ “8 4

integer function, and for each for some .3ß I œ I 4%3 8 4%

,Ñ N œ Ö4 À Ð I Ñ 8 × l\ÐI Ñl Ð I Ñ If then .% %%

. ? % . ?8 4 8 4 8 44 N

% % %

%

! "8%

-Ñ 8 _% Ò% Ä !

monotonically.Then if is the complement of N N% %

-

» » » – — Ÿ»" " "" "

"

% . ? % . ?. ?

%

. ? %

. ?

3

3 8 4 8 4 8 4 8 4

4 4

8 4

4−N 4−N

8 4 8 4 8 4

4−N

48

\ÐI Ñ \ÐI Ñ Ð I Ñ œ \ÐI Ñ Ð I ÑÐ I Ñ

Ÿ l\ÐI Ñl Ð I Ñ l\ÐI Ñl

Ÿ l\ÐI Ñl"

8

Ð I

%

%%

% % % %

%

%

% % %

%

%

-

-

%4Ñ

8Ð$Þ(Ñ

Ÿ Ö" $U× !ß"

8

%

%

Ò% Ä !

23

the above holding for sufficiently small. Thus exists and equals% .V' \ÐIÑ.' \ÐIÑ. ß è. completing the proof.

As is clear from the standard central limit theorem, expectation of r.v.'s does not scaleunder independent sums in the same way as some other “linear” quantities, such asstandard deviation. For this reason expectation will in general diverge in the formation ofintegrals (in which may be non-linear), while all other linear parameters converge to9 %Ð Ñdefine a limiting distribution. To avoid this inessential complication, we assumehenceforth that integrals involve -mean random variables.!

Definition 3.3: If is an r.v. or probability distribution, denotes the convolution of] ] ‡8

] 8 with itself times.

Recall that a distribution function is if for every and , thereJ , ß , + ß + !stable " # " #

exist constants and such that, + !

JÐ+ B , чJÐ+ B , Ñ œ JÐ+B ,ÑÞ Ð$Þ)Ñ" " # #

Note that stability implies infinite divisibility..

Proposition 3.4: If exists and is non-zero, then is stable,] œ \ÐIÑ Ð. Ñ ]' 9 .

6 œ Ð8 œ "ß #ß $ßá ÑÐ Ñ

Ð8 Ñ8

Ä!lim%

9 %

9 %

exists, and 6 ] œ ] Þ8‡8

Proof: Fix and let be a sequence such that . Consider a spectral -net8 Ö × % % .5 5" 8%5

Y Y Y% % %œ ÖI × 8ß4 4−N%% such that consists of with each element repeated times. For

5 58

5 œ "ß #ß $ßá ß let

] œ Ð8 Ñ \ÐI Ñ#5" 5 8 ß4

4−N

9 % "8 5

5

%

%

] œ Ð Ñ \ÐI Ñß#5 5 ß4

4−N

9 % "%5

5%

where indicates a product.8%5

Then

] œ ] ÞÐ Ñ

Ð8 Ñ#5 #5"

‡8 5

5

9 %

9 %

Letting , we conclude that5 Ä _

24

6 œÐ Ñ

Ð8 Ñ8

5Ä_

5

5lim

9 %

9 %

exists, and that 6 ] œ ] Þ è8‡8

Corollary 3.5: If of Proposition 3.4 has finite second moment then it is normal.]

Proof À The normals are the only stable distributions with finite second moment (see[GK]). è

The following theorem generalizes Liapounov's theorem (see, e.g. [Ch]) to infinitesums of random variables, and will be instrumental in subsequent limit theorems. Recallthat convergence in law of a net of random variables or distributions is denotedÖ\ ×3 3−M

by is the normal distribution with mean and variance \ Ê Q RÐ7ß Ñ 7 Þ3# #5 5

Theorem 3.6: For each , let be independent zero-mean r.v.'s on the% ! Ö\ ×%4 "Ÿ4ŸO%

same probability space and (or ). LeO _ O œ _ >% %%ÒÄ!

# X 5 X% % %%4 4 4$ $ # #

4œ l\ l ß œ \ ߊ ‹ Š ‹# # 5 5% % % %$ $ # #

4−O 4œO4 4œ ß œ Þ" "

% %

Then if the sum is asymptotically normal:#5 %

%%

%%

ÒÄ! 4−N

4!ß \!"

\ Ê RÐ!ß "ÑÞ Ð$Þ*Ñ5%

%%

"4

4Ä!

Proof: Let be an integer such that 5 Ÿ O Ð3Ñ% %

!Π!

4œ"

5

4$

4œ"

5

4#

Ä!

%

%

#

5

Ò%

%

%$#

!

and Ð33Ñ

25

! Š ‹!

4œ5 "

O

4#

4œ"

5

4#

Ä!

%

%

%

5

5

Ò

%

%

%!Þ Ð$Þ"!Ñ

The sum is asymptotically normal by and Liapounov's theorem:!4œ"

5

4

%

\ Ð3Ñ%

!!4œ"

5

4

4œ"

5

4#

Ä!

%

%

\

Ê RÐ!ß "ÑÞ

%

%

%Š ‹5"#

By the same holds forÐ33Ñ

X ´ Þ Ð$Þ""Ñ

\

%

%

%

!Š!

4œ"

5

4

4œ"

O

4#

%

%

5

"#

The remainder

Y ´ Ð$Þ"#Ñ

\

%

%

%

!Œ !4œ5 "

O

4

4œ"

O

4#

%

%

%

5

"#

satisfies . HenceX ÒÐY Ñ !% %

#

Ä!

U ´ X Y Ê RÐ!ß "ÑÞ è Ð$Þ"$Ñ% % %%Ä!

Theorem 3.7: If and , thenV V! !

_ _# $' ' Š ‹X . X .Ð\ Ñ. l\l . _

(!

_

\ÐIÑÐ. Ñ œ RÐ!ß @Ñß."#

where .@ œ Ð\ Ñ.V !

_ #' X .

Proof: Let for be a spectral -net, and . Let andY % . 5% % % % %œ ÖI × ! \ ´ \ÐI Ñ4 4−N 4 4 4

#%4 be as above. Then

26

#

5

X % X

5 % X

% ÒX .

X .

%

%

% %

% %

%

$

$

4 44 4$ $

4 44 4# #

Ä!

$

#

œ œ ! † œ ! Þ Ð$Þ"%Ñ

l\ l l\ l

Ð\ Ñ

l\l .

l\l .

! !Š ‹ Š ‹Œ Œ ! !

' Š ‹Š Š ‹ ‹'$ $ $

# # #

"#

Hence

( " Š ‹\ÐIÑÐ. Ñ œ \ œ † @ œ @ RÐ!ß "Ñ œ RÐ!ß @Ñà

\

. %

%

% 5

" " " "# # # #

"#

"#

lim lim% %

%

%

%

Ä! Ä!4

44

4

44#

!!

the second equality follows by Theorem 3.2, and the third by Theorem 3.6. è

§3.2 Background for Singular IntegralsNow we prove a result central to the singular integral theory in Chapters 4 and 5. A

singular integral is one where is singular. Since such integrals' \ÐIÑ Ð. Ñ Ð\ ÐIÑÑ9 . X #

fail to exist in general under the current definition, we make one which is more useful.

Definition 3.8: Let be a one parameter family of independent r.v.'s on , and \ÐIÑ ‘ .

a -finite measure on Let , and for each , let 5 ‘ . % Y _4 4œ"Þ J ÐIÑ œ ÐÒ!ßIÓÑ ! œ ÖI ×% %

be the ordered set of discontinuities of , a jump discontinuity of size being listed’ “JÐIÑ% 8

8 times. We define

W

( "‘ %

%\ÐIÑ Ð. Ñ ´ Ð Ñ \ÐI ÑÞ Ð$Þ"&Ñ9 . 9 %limÄ!

4œ"

_

4

Theorem 3.9: Suppose and are functions and9 % < %Ð Ñ Ð Ñ

<

9Ò X 9 . X < .

$

Ä!

$ # #

%!ß l\l Ð. Ñ _ß ! Ð\ Ñ Ð. Ñ ´ @ _Þ

W W( (Š ‹Then

W' \ Ð. Ñ œ RÐ!ß @ÑÞ< .

Proof: By the hypotheses,

9 % X

< % X

Ò

Ð Ñ l\ l

Ð Ñ Ð\ Ñ

O _

! Š ‹Š ‹!4

4$

$

44#

Ä!

%

%

%$#

.

Hence

27

! Œ ‹Š ‹!4

4$

44#

Ä!

X

X

Ò

l\ l

Ð\ Ñ

!ß Ð$Þ"'Ñ

%

%

%$#

and is asymptotically normal by Theorem 3.6; the variance of is! !4 4

4 4\ Ð Ñ \% %< %

< % X# #

44Ð Ñ Ð\ Ñ è!% , completing the proof.

Corollary 3.10: If and$6 5 !

W W( (Š ‹X . X .l\l Ð. Ñ _ß @ œ Ð\ ÑÐ. Ñ Á !ß$ 5 # #6

then

W( \Ð. Ñ œ RÐ!ß @ÑÞ. 6

§3.3 Distributions Under Symmetric Statistics

Given a spectral measure and a spectral -net , the operator with spectrum . . Y Y% % %Einduces a canonical ensemble under the conditions outlined in Chapter 2. For a net ofoperators , (with a function on ) we now consider asymptotics ofG œ 1ÐE Ñ 1% % ‘

corresponding value functions ; applications will be deferred until later.-%

Definition 3.11: Let denote the functions of on with two continuous derivativesK I" ‘

near , locally of bounded variation, such that is non-increasingB œ ! l1ÐIÑlÐ/ "Ñ"I "

for sufficiently large. For , those non-trivial spectral measures which are 0-freeI 1 − K"

and satisfy

! 1 ÐIÑ/ . ÐIÑ _ Ð$Þ"(Ñ(!

_8 I" .

for are denoted by . Those whose spectral functions have8 œ !ß "ß #ß $ V − V1ß 1ß" ".three continuous derivatives near 0 comprise W Þ1ß"

The monotonicity condition on implies the same forl1ÐIÑlÐ/ "Ñ"I "

l1ÐIÑlÐ/ "Ñ Þ"I "

The first and second conditions on guarantee existence of aÐ8 œ !Ñ − V. 1ß"

symmetric canonical ensemble, while the extra one on is required to controlW1ß"

asymptotic behavior of . Let denote the spectral measures for which has an- Y -% " %1ß

asymptotic distribution. Here and in Chapters 4 and 5 we show that .W § Y1ß 1ß" "

28

Proposition 3.12: Let and be non-decreasing and non-increasing,0Ð † Ñ 1Ð † Ñrespectively, with possibly infinite, and . Then1Ð+Ñ 0Ð+Ñ œ !

! Ÿ 1ÐBÑ.0ÐBÑ 1ÐBÑ.Ò0ÐBÑÓ Ÿ 1ÐBÑ.0ÐBÑß Ð$Þ")Ñ( ( (+ + +

, , Bw

where

B œ Ð ÖB À 0ÐBÑ   "×ß ,Ñßw min inf

, _ + , + , may be , and “ ” and “ ” may be replaced by “ ” and “ ”, respectively.

Proof: The proof is the same as that of Lemma 2.5 up to (2.20). We have

" " "( ( (5œ" 5œ" 5œ#W W W5 5 5

1ÐBÑ.0ÐBÑ   1ÐBÑ.Ò0ÐBÑÓ   1ÐBÑ.0ÐBÑß Ð$Þ"*Ñ

since the fact implies that0Ð+Ñ œ !

(W

55

1ÐBÑ.0ÐBÑ   1ÐB Ñ

for all , including . Equation (3.19) immediately gives (3.18) when . The5 5 œ " , œ _case of finite follows similarly, as do the replacements of “ ” and “ ” by “ ” and, + , +

“ ”. , è

Proposition 3.13: Let be of bounded variation and non-increasing for large ,l1ÐBÑl Bwith non-decreasing. Then0

( (” •+ +

, ,

1ÐBÑ. œ 1ÐBÑ.0ÐBÑ SÐ"Ñ Ð Ä !Ñß0ÐBÑ "

% %%

where may be infinite, and are allowed., + Ä + ß , Ä ,

Proof: Assume . Since has bounded variation, it suffices to assume is, _ 1 1monotone. If is non-increasing, the result follows from Lemma 2.5, the general case1following similarly. If we may assume that is non-increasing, since any, œ _ l1linterval in which this fails may be disposed of by the above. In this case Lemma 2.5again applies, completing the proof. è

29

Letting above, we obtain0ÐBÑ œ B

" (5œ "

+

,

’ “

’ “

+

,

%

%

1Ð5 Ñ œ 1ÐBÑ.B SÐ"ÑÞ Ð$Þ#!Ñ"

%%

Theorem 3.14: Let , and and one or both of the following hold:1 − K − W ß" ". 1ß

Ð3Ñ J Ð!Ñ œ J Ð!Ñ œ !à − Ww ww1ß. "

Ð33Ñ 1Ð!Ñ œ !à − V ß. 1ß"

where is the spectral function of . Let be the -discrete operator corresponding toJ E. %%

. >, and . Then the value function of satisfies- œ 1ÐE Ñ - . ÐG Ñ% % % %

È %%

- Ê RÐ!ß @Ñß7

%%Ä!

(3.21)

where

7 œ 1ÐIÑÐ/ "Ñ . ß @ œ 1ÐIÑ/ Ð/ "Ñ . Þ Ð$Þ##Ñ( (! !

_ _I " I I #" " ". .

Proof: Ð3Ñ 8 Let denote occupation numbers on the symmetric canonical ensemble over%4

E à% then

- œ 1ÐI Ñ8% % %" 4 4

and , where denotes the geometric r.v. with parameter . Let8 µ Ð/ Ñ Ð/ Ñ /%" " "

4 I I I# #%4

Ö\ÐIÑ× Ð/ Ñ%"

! I denote independent r.v.'s, and#

\ÐIÑ ´ \ÐIÑ Ð\ÐIÑÑÞ Ð$Þ#$Ñs X

Then

W

& &% %

W( (+ !

_ _$ $ $X . X X .Ðl1ÐIÑ\ÐIÑl ÑÐ. Ñ Ÿ l1 ÐIÑl Ð\ÐIÑ Ð\ÐIÑÑÑ Ð. Ñ Þ Ð$Þ#%Ñs

A calculation shows

X Xœ œÐ\ÐIÑ Ð\ÐIÑÑÑ œSÐ/ Ñ ÐI Ä _Ñ

SÐI Ñ ÐI Ä !Ñ$

I

$

"

.

Hence the right side of (3.24) is bounded by

30

- I Ð. Ñ - / l1 ÐIÑlÐ. Ñ ß Ð$Þ#&Ñ" #!

$ I $_

W

& &% %

W( ($

$

". .

where is chosen such that is not an atom of , for some$ $ . ! Ð3Ñ Ð33ÑO !ß JÐIÑ Ÿ OI ÐI Ÿ Ñ Ð333Ñ J I Ÿ$ $ $, and is continuous for . The second termis 0 by Theorem 3.2. For , and thusI Ÿ ß œ 4% %4

JÐI Ñ$ %4

I   ß4

O%4 Π%

"$

so that the first term is bounded by

lim lim lim ln% % %

$$

Ä! Ä! Ä!" "

I Ÿ

"

Ÿ

"- Ÿ - O 4 œ SÐ Ñ œ !Þ Ð$Þ#'Ñ4

O% % % %

%& " "% % %

4 45

"$

" "Œ % %Š ‹

Hence, the left side of (3.24) is 0, while

! Ð1ÐIÑ\ÐIÑÑ . œ 1 ÐIÑ Ð\ÐIÑÑ.s

œ 1 ÐIÑ/ Ð/ "Ñ . _Þ

W W

W

( (Š ‹(

! !

_ _# #

!

_# I I #

X . i .

." "

By Corollary 3.10 with and 5 œ 6 œ& "% # ß

W

"#(

!

_

1ÐIÑ\ÐIÑÐ. Ñ œ RÐ!ß @Ñß Ð$Þ#(Ñs .

with

@ œ . Þ1 ÐIÑ/

Ð/ "Ñ(!

_ # I

I #

"

".

Thus is asymptotically normal.-%To calculate the asymptotic expectation of , we assume initially that . Then- 1Ð!Ñ !%

XÐ- Ñ œ œ ß1ÐI Ñ 1ÐI Ñ

Ð/ "Ñ Ð/ "Ñ%

% %

" "$ $

" " " 4

4 4

I II Ÿ I

% %

% %

4 4

4 4

with chosen so that it is not an atom of , and the absolute value of the summand is$ .decreasing in for . By Proposition (3.13) the second sum on the right isI I Ÿ% %4 4 $

31

W( (– —$ $

" "

_ _

I I

1ÐIÑ JÐIÑ " 1ÐIÑ

/ " / ". œ . SÐ"Ñß Ð$Þ#)Ñ

% %.

while

" ( (– —I Ÿ

4

I! !

I I%

%

4

4$

%

"

$ $

" "

1ÐI Ñ

Ð/ "Ñœ . œ . 2Ð Ñß Ð$Þ#*Ñ

1ÐIÑ 0ÐIÑ " 1ÐIÑ

Ð/ "Ñ Ð/ "Ñ% %. %

where

l2Ð Ñl Ÿ . ß" l1ÐIÑl

Ð/ "Ñ% .

%(!

I Ð Ñ

I

w %

"

and . If is 0 in a neighborhood of 0 then (3.28) holdsI Ð Ñ ´ I À   " Jw w% inf š ›0ÐI Ñw

%

for . Otherwise, and$ %œ ! I Ð Ñ !w Ò% Ä !

l2Ð Ñl Ÿ .JÐIÑ Ÿ .JÐIÑ œ . ß- " - " - "

Ð/ "Ñ I I Ð Ñ% %

% % % %( ( (! ! !

I Ð Ñ I Ð Ñ JÐI Ð ÑÑ

I w"

w" " "% % %

"

for some and sufficiently small. The equality obtains as follows. Let be thrice- ! J%continuously differentiable for is an inverse of since I Ÿ Þ JÐ † Ñ I Ð † Ñß J‰I Ð Ñ œ( % %" "

Ð! Ÿ Ÿ JÐ ÑÑ J Ð!ß Ó% ( ( (note that is not generally invertible on , since it may be constanton intervals). Hence equality follows formally by the change of variables ; a%w œ JÐIÑrigorous argument involves the definition of the Stieltjes integral as a limit of sums overpartitions. Since JÐIÑ Ÿ OI ÐI − Ð!ß ÓÑß$ (

I Ð Ñ   Ð! Ÿ JÐ ÑÑÞO

" % % (%Œ

"$

Hence,

(!

JÐI Ð ÑÑ

"w

w"

"" "$ $# #

$ $

% " $O $O

I Ð Ñ # #. Ÿ JÐI Ð ÑÑ œ Þ Ð$Þ$!Ñ

%% % %

Combining (3.28-30),

X . % % %% %

Ð- Ñ œ . ! œ S Ð Ä !ÑÞ Ð$Þ$"Ñ" 1ÐIÑ 7

/ "% "( Š ‹ Š ‹

!

_

I " "

$ $

This calculation is much simpler if (and hence near zero). By1Ð!Ñ œ ! 1ÐIÑ œ SÐIÑ(3.27) and (3.31),

È È ÈŒ Œ % % X % X% %

- œ Ð- Ð- ÑÑ Ð- Ñ 7 7

% % % %

32

Ê 1ÐIÑ\ÐIÑÐ. Ñ ! œ RÐ!ß @Ñßs%Ä! !

_

W

"#( .

completing the proof of .Ð3Ñ

In this case , andÐ33Ñ 1ÐIÑ œ SÐIÑ ÐI Ä !Ñ

XÐl1ÐIÑ\ÐIÑl Ñ œ Ð$Þ$#Ñs SÐ/ Ñà ÐI Ä _ÑSÐ"Ñà ÐI Ä !Ñ

$ Iœ "

with the same bounds for . Hence asymptotic normality of follows byXÐ1 ÐIÑ\ ÐIÑÑ -s##

%

Corollary 3.10, with . By Proposition 3.13,5 œ "ß 6 œ "#

X .% %

%.

Ð- Ñ œ . œ . SÐ"Ñà1ÐIÑ JÐIÑ " 1ÐIÑ

/ " / "

Z Ð- Ñ œ . SÐ"Ñß" 1ÐIÑ/

Ð/ "Ñ

% " "

%

"

"

( (– —(! !

_ _

I I

!

_ I

I #

completing the proof. è

§3.4 Antisymmetric Distributions

Definition 3.15: The class is defined in the same way as (Definition 3.11),L K" "

without the condition on differentiability. For is without the 0-freedom1 − L ß X V" " "1ß 1ß

condition.

The relaxed conditions above still allow proof of central limit theorems for observablesunder antisymmetric statistics, since antisymmetric occupation numbers form a non-singular family. In this case is Bernoulli (see §2.2), with moments easily bounded by8%4

X XÐl8 l Ñ Ÿ 8 Ð/ "Ñ œ SÐ/ Ñ Ð5 œ !ß "ß #ßá Ñ Ð$Þ$$Ñs% %" "

4 45 I " I

5 Π% %4 4

where

8 ´ 8 Ð8 Ñ œ 8 / " Þs% % % %"

4 4 4 4I

"

X Š ‹%4

By Corollary 3.10, is asymptotically normal. By Proposition 3.13,- œ 1ÐI Ñ8% % %!

4 4

33

X .% %

%. %

Ð- Ñ œ . œ . SÐ"Ñ Ð$Þ$%Ñ1ÐIÑ JÐIÑ " 1ÐIÑ

/ " / "

Z Ð- Ñ œ . SÐ"Ñ Ð Ä !ÑÞ" 1ÐIÑ/

Ð/ "Ñ

% " "

%

"

"

( ( – — (! !

_ _

I I

!

_ I

I #

Applying Proposition 3.13 to (3.34), we have

Theorem 3.16: If and then1 − L − X ß" ". 1ß

È %%

- Ê RÐ!ß @Ñß7

%

where

7 œ . ß @ œ .1ÐIÑ 1ÐIÑ/

Ð/ "Ñ Ð/ "Ñ( (! !

_ _

I I #

I

" "

"

. ..

34

Chapter 4

Singular Central Limit Theorems, I

The integration theory of Chapter 3 deals with non-singular integrals in that, e.g.,W' X .Ð\ Ñ.# is finite. This section, from a probabilistic standpoint, is an illustration of the

singular theory for the family of geometric r.v.'s arising under symmetric statistics. Thenon-normality of these integrals, demonstrated in Chapter 5, is closely connected to the"infrared catastrophe" of quantum electrodynamics.

The most convenient formulation (with a view toward applications) is in the languageof limits rather than integrals. By Theorem 3.14, we may assume that .1Ð!Ñ !

§4.1 The Case JÐIÑ œ +I ß 1ÐIÑ ´ "#

With the assumptions of §3.3 still in force, this falls into the category of “weakest”violations of the hypotheses of Theorem 3.14. We have

I œ à Z Ð- Ñ œ +Ð Ð ÑÑ S ß Ð%Þ"Ñ4 "

+% %4

#Ê Œ %"9 %

%

where

9 %%

%Ð Ñ œ

l lÊ ln

(a definition which will hold henceforth); hence Theorem 3.14 cannot hold here. Theasymptotic distribution of , though normal, has a new nature in that, after normalization-%to standard variance and mean, is asymptotically dominated by occupation numbers-%8 I% %4 4 whose spectral values are arbitrarily small. Normality is in fact not universal inthis situation, as will be seen later.

Note that in all “singular” sums of r.v.'s to be considered, leading asymptotics will-%depend only on spectral behavior near the origin; parameters related to global spectraldensity will not contribute. We now consider a prototype.

Lemma 4.1: , , If and thenJÐIÑ œ +I ÐI   !Ñ 1ÐIÑ œ "#

V ´ Ð Ñ - Ê R !ß Þ Ð%Þ#Ñ$

+% %9 %

1

%" "ΠΠ#

# #

Proof: Assume without loss that . The characteristic function of is+ œ " V%

35

F X XV3V > 3> Ð Ñ-

3>

3>

5œ"

_ 5

Ð 53 Ð Ñ>Ñ

%% %

9 % 1

%"

9 % 1

%"

Ð>Ñ œ Ð/ Ñ œ / /

œ / ß" /

" /

Œ

Œ ÈÈ

Ð Ñ #

$ #

Ð Ñ #

$ #

Œ $ Ÿ

9 %

" %

" % 9 %

so that

ln ln lnF9 % 1

%"V

#

#5œ"

_ 5 Ð 53 Ð Ñ>Ñ

%Ð>Ñ œ " / " / Þ

3> Ð Ñ

$" ŸŠ ‹ Š ‹" % " % 9 %È È

Ð%Þ$ÑThe branch of the logarithm is determined by analytic continuation from the real axis for5 large. We rewrite (4.3) as

lnF9 % 1

V

#

%Ð>Ñ œ

3> Ð Ñ$%"#

" / Ð 5Ñ"Š Š ‹ ‹È5œ"

5

’ “ È"%

ln ln" % " %

" / Ð 5 3 Ð Ñ>Ñ"Š Š ‹ ‹È5œ"

Ð 53 Ð Ñ>Ñ

’ “ È"%

ln ln" % 9 % " % 9 %

5 Ð 5 3 Ð Ñ>Ñ" ŸŠ ‹È È5œ"

’ “"%ln ln" % " % 9 %

" / " /" ŸŠ ‹ Š ‹5œ "

_ 5 Ð 53 Ð Ñ>Ñ

’ “È È

"%

ln ln" % " % 9 %

œ 2Ð 5Ñ 2Ð 5 3 Ð Ñ>Ñ" ŸÈ È5œ"

’ “"%" % " % 9 %

Ð 5Ñ Ð 5 3 Ð Ñ>Ñ Ð%Þ%Ñ" ŸÈ È5œ"

’ “"%ln ln" % " % 9 %

36

6Ð 5Ñ 6Ð 5 3 Ð Ñ>Ñ ß" ŸÈ È5œ "

_

’ “"%

" % " % 9 %

where

2ÐBÑ œ à 6ÐBÑ œ Ð" / ÑÞ" /

Bln lnΠB

B

In the first sum ranges from to , so that%5 ! "

" ŸÈ È

Ÿ" È5œ"

5œ"

w

’ “

’ “

"

"

%

%

2Ð 5Ñ 2Ð 5 3 Ð Ñ>Ñ Ð%Þ&Ñ

œ 2 Ð 5ÑÐ3 Ð Ñ>Ñ Ð" SÐ Ð ÑÑÑ Ð Ä !ÑÞ

" % " % 9 %

" % 9 % 9 % %

Similarly

" ŸÈ È

Ÿ" È5œ "

_

5œ "

_w

’ “

’ “

"

"

%

%

6Ð 5Ñ Ð 5 3 Ð Ñ>Ñ Ð%Þ'Ñ

œ 6 Ð 5Ñ3 Ð Ñ> Ð" SÐ Ð ÑÑÑ Ð Æ !ÑÞ

" % " % 9 %

" % 9 % 9 % %

ln

Note that is bounded and monotone decreasing for ,2 Ð BÑ œ B − Ò!ß "Ów B"/

BÐ/ "Ñ"È "

"

ÈÈ"

"

ÈÈ

B

B

as is , for . Thus (3.20) applies to the sums in (4.5) and6 Ð BÑ œ B − Ò"ß_Ñw "

/ ""È "ÈB

(4.6), and (4.4) becomes

lnF " 9 % 9 %9 % 1

%" %V

#

#!

"w

%Ð>Ñ œ 2 Ð B Ñ.B SÐ"Ñ Ð" SÐ Ð ÑÑÑ3 Ð Ñ>

3> Ð Ñ "

$ Ÿ( È

6 Ð BÑ.B SÐ"Ñ Ð" SÐ Ð ÑÑÑ3 Ð Ñ>" Ÿ( È%

" 9 % 9 %"

_w

Ð 5Ñ Ð 5 3 Ð Ñ>Ñ Ð%Þ(Ñ" ŸÈ È5œ"

’ “"%ln ln" % " % 9 %

œ .B .B Ð" SÐ Ð ÑÑÑ 3> Ð Ñ 3 Ð Ñ> " "

$ / " B

9 % 1 9 %

%" % "9 %

#

#! !

_ "

B Ÿ( ( È"È

37

" Ð Ä !ÑÞ3 Ð Ñ>

5" Œ È5œ"

’ “"%ln

9 %

" %%

For small the arguments of all logarithms are near , so that branches are defined% ‘ ‚ §by analytic continuation.

For and , where is bounded.B − lBl Ÿ ß Ð" BÑ œ B Ð" BWÐBÑÑ WÐ † Ñ‚ " B# #ln #

Noting that

9 % 9 %

" % " %Ò

Ð Ñ> Ð Ñ>

5Ÿ ! Ð%Þ)ÑÈ È %Ä!

we have

" " " È È5œ" 5œ" 5œ"

# #

#

’ “ ’ “ ’ “" " "% % %

ln " œ Ð" 9Ð"ÑÑ3 Ð Ñ> 3 Ð Ñ> Ð Ñ >

5 5 # 5

9 % 9 % 9 %

" % " % " %

œ Ð" 9Ð"ÑÑ Ð%Þ*Ñ 3 Ð Ñ> " Ð Ñ > "

5 # 5

9 % 9 %

" % %"È È" " 5œ" 5œ"

# #

#

’ “ ’ “" "% %

œ # SÐ"Ñ SÐ"Ñ Ð" 9Ð"ÑÑ Ð Ä !ÑÞ 3 Ð Ñ> " Ð Ñ > "

#

9 % 9 %

" % % %" %%È – — – —

ÍÍÍÌ# #

#ln

Finally, replacing by we have’ “" "% %

" È5œ"

#

#

’ “"%ln " œ 9Ð"Ñ Ð Ä !Ñß

Ð Ñ> #3 Ð Ñ> >

5 #

9 % 9 %

" % "% "%

thus

lnF9 % 1 9 % 1 9 %

%" % " " "% "V

# # #

# # #%Ð>Ñ œ 9Ð"Ñ Ð%Þ"!Ñ

3> Ð Ñ 3 Ð Ñ> # #3 Ð Ñ> >

$ $ # Ÿ

œ 9Ð"Ñ Ð Ä !ÑÞ>

#

#

#"%

Since is the characteristic function of , the Lévy-Cramér convergence/ R !ß >#

#" Š ‹""#

theorem completes the proof. è

38

§4.2 The Case Ð1J Ñ Ð!Ñ !ß 1J Ð!Ñ œ !w w w

In this case the “leading behavior” of arises from its small terms and can be- I% %4

analyzed with use of the previous lemma. Recall that the convolution of distributionfunctions is defined by

ÐJ ‡J ÑÐBÑ œ J ÐB CÑ.J ÐCÑß Ð%Þ""Ñ" # " #(with the obvious extension to probability laws.

Lemma 4.2: Let be a collection of independent r.v.'s and be another.Ö\ × Ö\ ×3 3œ" 3œ"_ w _

3

Let and converge a.e. If for all thenW œ \ W œ \ J Ÿ J ÐBÑ Bß! !4 \ \w w

4 3 3w

J ÐBÑ Ÿ J ÐBÑÞW Ww

Proof: This follows by direct calculation for finite sums, and in the general case from thefact that almost sure implies distributional convergence. è

The sets and are defined in Definition 3.11.K W" "1ß

Theorem 4.3: Let and , with and the spectral function of .1 − K − W 1Ð!Ñ ! J" ". .1ß

Assume that and . ThenJ Ð!Ñ œ ! J Ð!Ñ œ #+ !w ww

9 %% "

Ð Ñ - Ê R !ß ß7 +1Ð!ÑŒ Œ % #

with given by (3.22).7

Proof: We prove this in three parts, first assuming , and then1 ´ "ß J Ð! Ñ !www

sequentially eliminating the second and first conditions.

I. Assume

J Ð! Ñ !ß 1ÐIÑ ´ "Þwww

Then

- œ 8 œ 8 ß% % %"4œ"

_

4

and since. − W ß1ß"

J ÐIÑ ! Ð! Ÿ I Ÿ Ñ Ð%Þ"#Ñwww -

for some . Let- !

39

J ÐIÑ œ ß Ð%Þ"$ÑJÐIÑà I ŸJÐ Ñà I  

- œ -- -

.- - - -% % be the spectral measure defined by and the corresponding totalJ Ð † Ñß 8 œ 8!

44

occupation number. Define

L ÐIÑ œ ß Ð%Þ"%ÑJ ÐIÑ +I J ÐIÑ

%

- -– — – —% %

#

and let be the ordered discontinuities of , the size of aY% % %‡ ‡

4 "Ÿ4_œ ÖI × L Ð † Ñ

discontinuity being its multiplicity in . Let be the discontinuitiesY Y% % %‡ + +

4 "Ÿ4_œ ÖI ×

of , so that’ “+I#

%

I œ Þ Ð%Þ"&Ñ4

+%4+ Ê %

Let and be self-adjoint operators in Hilbert space with spectra and ,E E% % % %‡ + ‡ +Y Y

respectively, again with multiplicity. Note that

! Ÿ L ÐIÑ Ÿ " ÐI − Ñß Ð%Þ"'Ñ+I– —#

%‘%

since for ,+ß , − ‘

! Ÿ Ò+ ,Ó ÖÒ+Ó Ò,Ó× Ÿ "Þ

Furthermore, if , then, for ,I I I I I% %% %Ð4"Ñ Ð4"ч + + ‡

4 4

” •+I  4ß L ÐIÑ 4 "ß

#

%%

contradicting (4.16). Thus,

I Ÿ I Ð# Ÿ 4 _ÑÞ% %Ð4"Ñ 4‡ + (4.17)

Also, by (4.16),

I Ÿ I Ð" Ÿ 4 _ÑÞ Ð%Þ")Ñ% %4 4+ ‡

Let and be the symmetric occupation number r.v.'sÖ8 × Ö8 ×% %4‡ +

"Ÿ4_ "Ÿ4_4

corresponding to and andE E ß% %‡ +

8 œ 8 ß 8 œ 8 Ð%Þ"*Ñ% % % %‡ ‡ + +

4œ" 4œ"

_ _

4 4" "total number r.v.'s. By Lemma 4.1 and the definition of E ß%

+

40

U ´ 8 Ê RÐ!ß "ÑÞ Ð%Þ#!ÑÐ Ñ +

+ $% %%

+ +#

# Ä!

"9 % 1

%"È Note also that by (4.15),

X i X"9 % " 9 %

" 9 % "

" "

%Ò

ŸÈ œ Š ‹ È

Š ‹ Š ‹È ÈŒ

Ð Ñ Ð Ñ

+8 œ Ð8 Ñ Ð8 Ñ

+

œ Ð Ñ "

+

Ð Ñ

Ð Ñ " Ð Ñ "

œ S !ß"

l l

% % %

%

% %

" "+ + #

## #

"

# #+

+ +

# #

exp

exp exp

ln % Ä !

so that

[ ´ 8 Ê ß Ð%Þ#"ÑÐ Ñ

+% %

%

+ +"

Ä!!

"9 %$È

with the point mass at 0. If we define$!

8 œ 8 8 ß% % %+ß" +

"

then (4.21) and (4.20) imply

U ´ U [ œ 8 Ê RÐ!ß "ÑÞ Ð%Þ##ÑÐ Ñ +

K $% % % %%

+ß" + + +ß"#

# Ä!

"9 % 1

%"È The d.f. of is8%4

+

J ÐBÑ œ Ð8 œ 5Ñ œ " / / œ " / ÐB − Ñß8 4

5œ! 5œ!

ÒBÓ ÒBÓ I 5I ÒB"ÓI

%% % %

4+ 4 4 4

+ + +" "Œ c ‘%" " "

which is monotonic in E . By (4.17) and (4.18),%4+

J ÐBÑ Ÿ J ÐBÑ Ð4   #Ñ Ð%Þ#$Ñ

J ÐBÑ Ÿ J ÐBÑ Ð4   "Ñ Þ

8 8

8 8

% %

% %

Ð4"ч

4+

4 4+ ‡

We decompose

Y Y Y% % %‡ ‡M ‡MMœ ß

where the set corresponds to the discontinuities in and to those inY Y% %%‡M ‡MMJ ÐIÑ’ “-

’ “+I J ÐIÑ ‡M ‡MM ‡# -

% % % %; the intersection of and is listed twice in . LetY Y Y

41

W œ Ö4 À I − ×ß W œ Ö4 À I − ×ß Ð%Þ#%Ñ% % % % % %M ‡ ‡M MM ‡ ‡MM

4 4Y Y

and define

8 œ 8 ß 8 œ 8 Þ Ð%Þ#&Ñ% % % %‡M ‡ ‡MM ‡

4−W 4−W

4 4" "% %M MM

Note that is the total occupation number corresponding to the d.f.8 8% %‡M ‡MMŠ ‹

J ÐIÑ œ J ÐIÑ œ Ð+I J ÐIÑÑ ÞJ ÐIÑ "

% %

--‡M ‡MM #

% %Œ

By their definitions, and , are identically distributed.8 8% %- ‡M

By (4.12-13), is monotone non-decreasing. Since ,J Ð † Ñ − W‡MM"ß. "

J − W ߇MM""

and

ÐJ Ñ Ð! Ñ œ ÐJ Ñ Ð! Ñ œ !Þ Ð%Þ#'чMM w ‡MM ww

By Theorem 3.14 and the aboveß

U ´ 8 Ê RÐ!ß "Ñß@

7% %‡MM ‡MM

‡MM

‡MMÊ %

%

where

7 œ .Ð+I J ÐIÑÑß"

/ "

@ œ .Ð+I J ÐIÑÑÞ/

Ð/ "Ñ

‡MM #

!

_

I

‡MM #

!

_ I

I #

((

"-

"

"-

The r.v.

8 œ 8 Ð%Þ#(Ñ% %+ß" +

4œ#

_

4"has term while the term of in (4.19) is . Thus by (4.23) the d.f. of the6 8 ß 6 8 8>2 + >2 ‡ ‡

Ð6"Ñ 6% %

6 8 8>2 +ß" ‡6 term of is bounded from above by that of the corresponding term in . Hence% %

by Lemma 4.2, letting denote the d.f. of ,J ÐBÑ 88

J ÐBÑ Ÿ J ÐBÑß8 8% %‡ +ß"

and similarly, by (4.24),

42

J ÐBÑ Ÿ J ÐBÑÞ8 8% %+ ‡

Since these inequalities are preserved under renormalization of expectations andvariances, we have by (4.20)

J ÐBÑ Ÿ J ÐBÑà J ÐBÑ Ÿ J ÐBÑß Ð%Þ#)ÑU U U U% % % %+ ‡ ‡ +ß"

where

U œ 8 ÞÐIÑ +

+ $% %‡ ‡

#

#

"9 1

%"È Thus by (4.28), and (4.22),

J ÐBÑ Ÿ J ÐBÑ Ÿ J ÐBÑ J ÐBÑß Ð%Þ#*ÑU U U Ä!RÐ!ß"Ñ% % %

+ ‡ +ß" Ò%

and by (4.29),

U Ê RÐ!ß "ÑÞ Ð%Þ$!Ñ%‡

But by the definition of ,U%‡

U œ 8 8 Ð%Þ$"ÑÐ Ñ + $ 7 Ð Ñ 7

+ +$

œ 8 U ÞÐ Ñ + $ 7 @ Ð Ñ

+ +$

% %

% %

‡ ‡M ‡MM# # ‡MM ‡MM

#

‡M ‡MM# # ‡MM

#

‡MM

"9 % 1 " "9 %

%" %

"9 % 1 " " 9 %

%" %

È È È È È

The coefficient of approaches 0 and so the second term converges in law to . ThusU%‡MM

!$by (4.31),(4.30), the independence of and , and the identity of and ,8 8 8 8% % % %

-‡M ‡MM ‡M

"9 % 1 "

%"

Ð Ñ + $ 7

+8 Ê RÐ!ß "ÑÞ

$È %-

%

# # ‡MM

# Ä!

Now let

8 œ 8 8 Þ Ð%Þ$#Ñ% %%-

Using by now standard arguments, we conclude

Ê % -

- %-

@ Ð Ñ8 Ð Ñ Ê RÐ!ß "Ñß Ð%Þ$$Ñ

7 Ð Ñ

%

where

7 Ð Ñ œ .JÐIÑà @ Ð Ñ œ .JÐIÑÞ" /

/ " Ð/ "Ñ

_ _

I I #

I

- -( (- -

" "

"

Equation (4.32) implies

43

"9 % "9 % 1 "

% %"-

"9 % 1-

% % %"

Ð Ñ 7 Ð Ñ + $ 7

+ +8 œ 8 Ð Ñ Ð%Þ$%Ñ

$

8 Ð Ñ ÞÐ Ñ 7 7 +

+ $ #

È È È

% %-

%

# # ‡MM

#

‡MM #

By the definitions,

7 7 œ .JÐIÑ œ 7 Ð Ñß Ð%Þ$&Ñ+ "

$ / "‡MM

#

# I

_1

"-(

-"

since

(!

_

I ##

#" +

/ " $.Ð+I Ñ œ Þ

"

1

"

Because , the last term in (4.34) converges to by (4.33) and (4.35). On the9 %

%

Ð Ñ!È Ä ! $

other hand, the first term on the right of (4.34) converges to the standard normal, so that

"9 %

%

Ð Ñ 7

+8 Ê RÐ!ß "Ñ Ð%Þ$'ÑÈ %

%Ä!

completing part I.

II. We now eliminate the restriction on and assume only that . InJ Ð!Ñ 1ÐIÑ ´ "www

this case there clearly exists a spectral function corresponding to a distributionJ Ð † Ñ!

. ‘! !"ß− W Ð"Ñ JÐIÑ J ÐIÑ I − Ð#Ñ" such that is monotone non-decreasing for ,

ÐJ Ñ Ð! Ñ œ !ß Ð$Ñ ÐJ Ñ Ð! Ñ œ + Ð%Ñ ÐJ Ñ Ð! Ñ !! w ! ww ! www , and . The construction of such aspectral function involves finding one with sufficiently negative third derivative which isconstant for all values of larger than some sufficiently small number . TheI I !"

function satisfies the hypotheses of Part I.J!

Analogously to (4.14), define

J œ Þ Ð%Þ$(ÑJ ÐIÑ JÐIÑ J ÐIÑ

%‡

! !– — – —% %

Let be the ordered set of discontinuities of and Y Y Y% % % % %‡ ‡ ‡

4 "Ÿ4_ 4 "Ÿ4_œ ÖI × œ ÖI ×

that of . As before’ “JÐIÑ%

! Ÿ J ÐIÑ Ÿ " ÐI − ÑßJÐIÑ– —%

‘%‡

so that

44

I Ÿ I à I Ÿ I Þ Ð%Þ$)Ñ% %% %4 44 Ð4"ч ‡

Let

8 œ 8 ß 8 œ 8 Ð%Þ$*Ñ% % % %" "4œ" 4œ"

_ _

4‡ ‡

4

be the total symmetric number of r.v.'s obtained from and , respectively, andJÐIÑ J ÐIч

8 œ 8 8% % %"

". As before we conclude from (4.38) that

J ÐBÑ Ÿ J ÐBÑà J ÐBÑ Ÿ J ÐBÑß Ð%Þ%!ÑV V V V% % % %‡ ‡ "

where

V œ 8 ßÐ Ñ 7

+% %

9 % "

%È with and corresponding similarly to and , respectively.V V 8 8% % % %

‡ " ‡ "

We make the decomposition , where corresponds toY Y Y Y% % % %‡ ‡M ‡MM ‡Mœ

discontinuities in and to those in . Let and be defined’ “ ’ “J ÐIÑ JÐIÑJ ÐIчMM ‡M ‡MM! !

% %% % %Y 8 8

analogously to (4.25). By Theorem 3.14,

Ê %

%@8 Ê RÐ!ß "Ñß Ð%Þ%"Ñ

7‡MM

‡MM‡MM

%

with

@ œ .ÐJÐIÑ J ÐIÑÑà Ð%Þ%#Ñ/

Ð/ "Ñ

7 œ .ÐJÐIÑ J ÐIÑÑÞ"

/ "

‡MM !

!

_ I

I #

‡MM !

!

_

I

((

"

"

"

By the conclusion of Part I,

"9 %

%

Ð Ñ 7

+8 Ê RÐ!ß "ÑßÈ %‡M

‡M

where

7 œ .J ÐIÑÞ"

/ "‡M !

!

_

I( "

Hence

V Ê RÐ!ß "ÑÞ Ð%Þ%$Ñ%‡

As in (4.21),

45

[ ´ 8 Ê Þ Ð%Þ%%ÑÐ Ñ

+% %

9 % "$È " !

Since is continuous, a net of probability distributions which converges toJ Ð † ÑRÐ!ß"Ñ

J Ð † Ñ BRÐ!ß"Ñ pointwise does so uniformly in . Thus by (4.44),

= ´ lJ ÐBÑ J ÐBÑ l !ß%‘ %

supB−

RÐ!ß"Ñ VÄ!%

‡ Ò

and

lJ ‡J ÐBÑ J ‡J ÐBÑ œ ÐJ ÐB CÑ J ÐB CÑÑ.J ÐCÑV [ [ V [RÐ!ß"Ñ RÐ!ß"Ñ_

_

% %% % %‡ ‡| (

Ð%Þ%&Ñ

Ÿ = .J ÐCÑ œ = !Þ % %(_

_

[%Ò% Ä !

Also by (4.44),

J ‡J ÐBÑ J ÐBÑßRÐ!ß"Ñ RÐ!ß"Ñ[%Ò% Ä !

so that

J ‡J ÐBÑ J ÐBÑÞ Ð%Þ%'ÑV [Ä!

RÐ!ß"Ñ% %‡ Ò

%

Thus Lemma 4.2, (4.44), (4.40), and (4.43) imply

J ‡J ÐBÑ Ÿ J ‡J ÐBÑ œ J ÐBÑ Ÿ J ÐBÑ J ÐBÑ ÐB − ÑßV [ [ V VVÄ!

RÐ!ß"Ñ% %% % %%‡ ‡" Ò ‘

%

so that, with (4.46), we have

V Ê RÐ!ß "ÑÞ Ð%Þ%(Ñ%

III. In the general case we may assume , by Theorem 3.14. Then1Ð!Ñ Á !

- œ 1Ð!Ñ8 Ð- 1Ð!Ñ8 ÑÞ% % % %

By Theorem 3.14 (with a slight modification if the integrand of below is not7"

asymptotically monotone),

Ê Š ‹%

%@- 1Ð!Ñ 8 Ê RÐ!ß "Ñß

7

"

"

Ä!% %

%

where

7 œ . à @ œ . ÞÐ1ÐIÑ 1Ð!ÑÑ Ð1ÐIÑ 1Ð!ÑÑ /

/ " Ð/ "Ñ" "

! !

_ _

I I #

# I( (" "

"

. .

46

Together with the result of Part II, this gives

È %% "

- Ê R !ß ß7 1Ð!Ñ+

% #

completing the proof. è

47

Chapter 5

Singular Central Limit Theorems, II

This chapter deals with the most singular families of geometric r.v.'s arising underour hypotheses, those from spectral measures with non-vanishing density near 0. Thesingularity of this situation fundamentally changes both the results and the proof of theassociated limit theorem. The normal distributions of integrals appearing in previouscases are now replaced by an extreme value distribution whose parameters are entirelydetermined by spectral density at 0. This will be shown in the prototypical case. ÐIÑ œ .JÐIÑ œ .Iß. and in general through an approach like that of Chapter 4. Theapproximately linear behavior of at 0 will be exploited to decompose into twoJÐIÑ -%independent r.v.'s, to the first of which the prototypical case will apply, and to the secondTheorem 4.3.

The is that with d.f. . The normalized distribution,extreme value distribution //B

with zero mean and unit variance, has d.f.

expŒ / ß Ð&Þ"Ñ BŠ ‹1È'

#

where is Euler's constant. For applications of this distribution to the# œ Þ&((ástatistics of extremes, see [G].

§5.1 Asymptotics Under Uniform Spectral DensityIn this section and are defined as before. We will require the following- ß ß 1% .

simple lemma.

Lemma 5.1: Let be a net of probability d.f.'s, andÖJ Ð † Ñ×% %!

J ÐBÑ JÐBÑ ÐB − Ñß Ð&Þ#Ñ%%Ò ‘Ä!

where is a continuous d.f. If is any net of d.f.'s, thenJÐ † Ñ ÖK Ð † Ñ×% %!

lim sup lim sup% %

% % %Ä! Ä!

J ÐBÑ ‡K ÐBÑ œ JÐBÑ ‡K ÐBÑ ÐB − Ñà Ð&Þ$Ñ ‘

(5.3) holds as well for the lim inf.

Lemma 5.2: If for some , and thenJÐIÑ œ ,I , ! 1ÐIÑ ´ "ß

48

V ´ - Ê / Þ,

% %

"%

%%

% ¹ Š ‹¹ln ,

Ä!

/B,"

Proof: We may assume . The logarithm of the ch.f. of is, œ " V%

ln ln lnln

F X"%

"V

3> V

4œ"

_ 4

43 >%

%"%"Ð>Ñ œ / œ 3> ß Ð&Þ%Ñ

Ð Ñ " /

" / "Š ‹%"%

"% %

lnÐ Ñ

where the branch of each term is principal for , and determined by analytic> !continuation from elsewhere. Fixing > ! >ß

lnln

F"%

"V%Ð>Ñ œ

3> Ð Ñ

Ð Ð" / Ñ Ð 4Ñ Ð" / Ñ Ð 4 3> ÑÑ

Ð Ð 4Ñ Ð 4 3> ÑÑ

Ð" / Ñ " /

"

"" Œ Œ

4œ"

4 43>

4œ"

4œ "

4 Ð 43

’ “

’ “

’ “

"

"

"

"%

"%

"%

ln ln ln ln

ln ln

ln ln

"% "% %

"% "% %

"% "% %

"% "% %

>ÑÞ

Ð&Þ&Ñ

Let

2ÐBÑ œ ß 6ÐBÑ œ Ð" / Ñß" /

Bln ln B

B

with branches as before. The function is analytic at all points within unit distance2Ð † Ñof the real axis interval and is analytic at all points within distance ofÒ!ß "Óß 6Ð † Ñ (

)

Ò!ß_Ñ H H > Á !. Denote these regions of analyticity by and , respectively. Let and2 6

l >l B − Ò!ß_Ñ% "# . If , then

6ÐB 3 >Ñ œ 6ÐBÑ 3 >6 ÐBÑ VÐBÑß% % w

where

» » » » » »Š ‹Š ‹ Š ‹

Š ‹ Š ‹Œ VÐBÑ " "

Ð3 >Ñ 6 ÐBÑ 3 > 3 >œ œ

S

% % %# w

"/"/

Ð >Ñ Ð >Ñ/ " / "

"/ /Ð/ "Ñ / "

#

ln ÐB3 >Ñ

B

# #

B B

3 > 3 >"

B B%

% %

% %

Ð&Þ'Ñ

49

Ÿ SÐ"ÑÞ

SŒŒ Š ‹/ "/ "

#

Ð >Ñ/ "

3 >

B

#

B

%

%

The term satisfiesSŠ ‹/ "/ "

3 >

B

%

SŸ Gß

Š ‹Š ‹

/ "/ "

/ "/ "

3 >

B

3 >

B

%

%

for some and sufficiently small values of the argument, andG !

SÐ"Ñ œ / " "

Ð >Ñ >

3 >

#

%

% %(5.7)

remains bounded for all , uniformly in (in fact independently of ). Thus if% ! B − H B6

> Á ! B B − H is fixed, is bounded uniformly in for and sufficiently small, andVÐBÑÐ3 >Ñ 6 ÐBÑ 6% # w %

"

Ÿ Ÿ" "4œ "

_ 4 43 >

4œ " 4œ "

_ _w

4

’ “

’ “ ’ “

"

" "

"%

"% "%

Ð Ð" / Ñ Ð" / ÑÑ Ð&Þ)Ñ

œ 3 >6 Ð 4Ñ Ð" SÐ ÑÑ œ Ð" SÐ ÑÑ Ð Ä !Ñß3 >

/ "

ln ln"% "% %

"%% "% % % %

%

which clearly also holds if .> œ !Similarly, one can show that for ,> − ‘

"

Ÿ"4œ"

4 43 >

4œ"4

’ “

’ “

"

"

"%

"%

Ð Ð" / Ñ Ð 4Ñ Ð" / Ñ Ð 4 3 >ÑÑ Ð&Þ*Ñ

œ 3 > Ð" SÐ ÑÑ Ð Ä !ÑÞ" "

/ " 4

ln ln ln ln"% "% %

"%

"% "% %

% % %"%

The summands in the last terms of (5.8) and (5.9) are decreasing in , and Proposition"%43.13 implies

"4œ "

_

4"

’ “""%

3 > 3>

/ "œ Ð" / Ñ 9Ð"Ñ Ð&Þ"!Ñ

%

""%ln

and

50

3 > œ Ð" / Ñ 9Ð"Ñ Ð Ä !ÑÞ Ð&Þ""Ñ" " 3>

/ " 4% %

"% "" 4œ"

4"

’ “""%

"%ln

Finally,

" $ $ – — – —

4œ" 4œ" 4œ"

’ “ ’ “ ’ “" " ""% "% "%

Ð Ð 4ÑÑ Ð 4 3 >Ñ œ 4 4 Ð&Þ"#Ñ3>

œ " " " ß" " 3> 3>

ln ln ln ln

ln ln ln

"% "% %"

> > >"% "% " "

where we continue our convention on branches. Stirling's expansion gives

ln ln ln> 1ÐDÑ œ D D D Ð# Ñ S ß Ð&Þ"$Ñ" " "

# # lDl uniformly for arg . ThusD Ÿ ? 1

ln ln ln> > >" "% "% " – — – — " " "3> " " 3>

œ " " "3> " " " " 3>

#ln ln ln>

" "% "% "% " – — – — – — Ÿ

" SÐ Ñ Ð&Þ"%Ñ3> " 3> 3>

œ " " SÐ Ñ3> " " 3> "

#

" "% " "%

> %" "% " "%

ln

ln

– — – — – —

"

#

Ð Ñ SÐ Ñ3> 3>

" ""% %ln

œ " Ð Ñ SÐ Ñ Ð Ä !ÑÞ3> 3>

ln ln> "% % %" "

Combining (5.5) and (5.8-14)

ln lnÐ Ð>ÑÑ œ " 9Ð"Ñß3>

F >"

V%

51

so that

F > Ò >" "

VÄ!%

Ð>Ñ œ " Ð" 9Ð"ÑÑ " Þ Ð&Þ"&Ñ3> 3> %

The right side is the Fourier transform of the distribution with d.f. , and the proof is// B"

completed by the Lévy-Cramér convergence theorem. è

§5.2 Asymptotics for Non-Vanishing Densities Near Zero

Theorem 5.3: If , and , then. − W J Ð!Ñ œ , !ß 1Ð!Ñ !1ßw

"

V ´ - Ê /7Ð Ñ

% %%

%%

% Ä!

/

B1Ð!Ñ,"

where

7Ð Ñ œ 7 Ð&Þ"'Ñ1Ð!Ñ,

,

7 œ .ÐJÐIÑ ,IÑ .JÐIÑÞ1Ð!Ñ 1ÐIÑ 1Ð!Ñ

/ " / "

%"

"%» »( (

ln ,

! !

_ _

I I" "

Proof: I. We assume initially that and1ÐIÑ ´ "

J ÐIÑ ! Ð! Ÿ I Ÿ Ñ Ð&Þ"(Ñww -

for a . Let be as in 4.13, and- ! J Ð!Ñ œ +ß J ÐIÑww -

L œ Þ Ð&Þ")ÑJ ÐIÑ ,I J ÐIÑ

%

- -– — – —% %

Let be as in Theorem 4.3, and be the discontinuitiesY Y% % % %‡ ‡ , ,

4 "Ÿ4_ "Ÿ4_4œ ÖI × œ ÖI ×

of i.e., .’ “,I4, 4

,% %%ß I œ

We have

! Ÿ L ÐIÑ Ÿ " ÐI − Ñ Ð&Þ"*Ñ,I– —%

‘%

and

I Ÿ I Ð" Ÿ 4 _Ñß I Ÿ I Ð# Ÿ 4 _ÑÞ Ð&Þ#!Ñ% % % %4 4 Ð4"Ñ 4, ‡ ‡ ,

52

Let and and correspond as before to and (seeÖ8 × Ö8 × ß 8 8% % % % %%4‡ , ‡ , ‡ ,

"Ÿ4_ "Ÿ4_4 Y Y

4.19). By the lemma,

U ´ 8 Ê / Þ Ð&Þ#"Ñ,

% %

"%

%

, , /,

Ä! Š ‹ln

"%

B,"

Let

8 œ 8 8 à 8 œ 8 8 à% % % % % %,ß4 , , ‡ß4 ‡ ‡

4 4

8 œ 8 à 8 œ 8 Ð !ß 4 œ #ß $ß %áÑÞ Ð&Þ##Ñ%

% %%4

, , ‡ ‡

5œ" 5œ"

4 4

5 54

v v

" " %

The d.f. of is%8%4,

J ÐBÑ œ " / " / Ð&Þ#$Ñ8"

Ä!

B%

%% "

4,

B 4,

4,

"

%

’ “Ò

and thus

[ ´ 8 Ê " / Þ Ð&Þ#%Ñ% %%

4 4, , B

Ä!%

"4,

We subdivide as in Theorem 4.3 into and with and defined asY Y Y% % % % %‡ ‡M ‡MM M MMW W

before. Since

,I J ÐIÑ

J ÐIÑ!ß

-

-ÒIÄ!

for there is an such that if , then4 − ! Ÿ % % %w

– —,I J ÐI Ñœ !ß

% %-

w w4 4‡M ‡M

w%

since

I œ I À   4 ÞJ ÐI Ñ

%

-

4‡M w

w

inf Ÿ%

Thus if is sufficiently small and is fixed% 5

I − Ð4 Ÿ 5 − ÑÞ Ð&Þ#&Ñ% %4‡ ‡MY

Since and by (5.17), there exist and such that if J Ð!Ñ œ , I ! P ! I Ÿ I ßwQ Q

,I PI Ÿ J ÐIÑ Ÿ ,IÞ Ð&Þ#'Ñ# -

Let , and be sufficiently small that equation (5.25) holds for some and4 − Ð3Ñ 5   4 %

53

Ð33Ñ I Ÿ I . Then%4‡

Q

I œ I œ I À   4 ÞJ ÐI Ñ

% %

-

4 4‡ ‡M w

w

inf Ÿ%

Together with (5.26) this implies

I œ SÐ Ñ Ð Ä !ÑÞ Ð&Þ#(Ñ4

,%4‡ #%

% %

Thus, as in (5.23),

J ÐBÑ œ " / " / Þ Ð&Þ#)Ñ%" %

%8

" SÐ Ñ

Ä!

B%

%% "

4‡

B 4,

# 4,

’ “Š ‹Ò

Let denote the exponential distribution with distribution function andW !Ð Ñ " / B!

W W W W" " "4 ´ ‡ ‡á‡ Þ, , ,

# 4 Then by (5.24) and (5.28)

[ ´ 8 Ê à [ ´ 8 Ê Ð4 − ÑÞ Ð&Þ#*Ñ% % % %%4

, , 4 ‡ ‡ 4

4 Ä! Ä!4

v v v v% W % W %4

Let

Q% %,ß4 ,ß4œ 8 ß

7Ð Ñ%

%

% 8 œ 8 8 ß 8 œ 8 8 ß% % % %

% %

,ß4 , , ‡ß4 ‡ ‡

4 4

v v

v v

and

U œ 8 ß U œ 8 Þ Ð&Þ$!Ñ7Ð Ñ 7Ð Ñ

% % % %,ß4 ,ß4 ‡ß4 ‡ß4

v v v v

% %% %

% % By (5.20)

J ÐBÑ Ÿ J ÐBÑ Ð !ß B − ß 4 œ #ß $ßá Ñß8 8% %Ð4"ч

4, % ‘

and hence

J ÐBÑ Ÿ J ÐBÑ Ð5 œ 4 " ß Ð&Þ$"Ñ8 8% %‡ß4 ,ß5

v v )

so that

54

J ÐBÑ Ÿ J ÐBÑ Ð5 œ 4 "ÑÞ Ð&Þ$#ÑU U% %

‡ß4 ,ß5 v v

Let

U œ 8 Þ Ð&Þ$$Ñ,

% %

"%

‡ ‡ ,%

"% ¹ Š ‹¹ln

Thus by Lemma 5.1

lim sup lim sup lim sup% % %Ä! Ä! Ä!

U [U

HU

J ÐBÑ œ J ‡J ÐBÑ œ J ‡J ÐBÑ%

% % %

‡ ‡

4 ‡ß44

‡ß4v v v

Ÿ J ‡J ÐBÑ œ J ‡J ÐBÑß Ð&Þ$%Ñlim sup lim sup% %Ä! Ä!

HU U[

4,ß5 ,ß5,ß4% %%

v v v

œ J ÐBÑ ÐB − − 5 œ 4 "Ñßlim sup%Ä!

U%,ß5 ‘ , j ,

where we have also used (5.29), (5.31), (5.32), Lemma 4.2, and (5.30).

We have , and thusU ‡[ œ U% %%,ß4 , ,

4

F F F % ‘ U

[ U%

% %,ß4 4, , vÐ>Ñ Ð>Ñ œ Ð>Ñ Ð !ß > − ß 4 − ÑÞ Ð&Þ$&Ñ

By (5.21) and (5.24),

F Ò > F Ò ‘ " "U [

"

% %, ,

4Ð>Ñ " à Ð>Ñ " Ð> − 4 −

,3> ,3>% %Ä ! Ä ! , ),

the right sides being characteristic functions of and , respectively. Thus/ 4/ B,"

WŠ ‹",

F Ò > F" "U Ä! ,

4

%,ß4Ð>Ñ " " ´ Ð>ÑÞ Ð&Þ$'Ñ

,3> ,3>

4% Since is continuous at 0,F,

4Ð † Ñ

J J ÐBÑU Ä! ,4

%,ß4 Ò

%

where

(_

_3B>

, ,4 4/ .J ÐBÑ œ Ð>ÑßF

and

55

lim sup%Ä!

U ,4J ÐBÑ Ÿ J ÐBÑ ÐB − ß 4 − ÑÞ Ð&Þ$(Ñ

%,ß4 ‘

Thus by (5.34) and (5.37)

lim sup%Ä!

U ,4J ÐBÑ Ÿ J ÐBÑ Ð4 − ÑÞ Ð&Þ$)Ñ

%‡

On the other hand, by (5.36),

J ÐBÑ / Þ,4

4Ä_

/Ò B,"

Since the left side of (5.38) is independent of ,4

lim sup%Ä!

U/J ÐBÑ Ÿ / Þ Ð&Þ$*Ñ

%

"

‡

B,

By (5.20),

J ÐBÑ Ÿ J ÐBÑ Ð !ß B − ß 4 − Ñß Ð&Þ%!Ñ8 8% %4,

4‡ % ‘

so that

J ÐBÑ Ÿ J ÐBÑÞ Ð&Þ%"ÑU U% %, ‡

Combining (5.21), (5.41) and (5.39) yields

/ œ J ÐBÑ Ÿ J ÐBÑ Ÿ J ÐBÑ Ÿ // /

Ä! Ä!U U U

Ä!

B B, ,, ‡ ‡

" "

% % %lim lim inf lim sup% % %

so that

J ÐBÑ / ÐB − Ñ Ð&Þ%#ÑUÄ!

/%

"

‡

B,Ò ‘%

and

U Ê / Þ Ð&Þ%$Ñ%‡ / B,

"

We now form

8 œ 8 8 ß Ð&Þ%%Ñ% % %‡ ‡M ‡MM

where and are defined as in (4.25). By arguments identical to those following8 8% %‡M ‡MM

equation (4.26), is identically distributed with the occupation number8 8 ß% %-‡M

corresponding to . We have analogously (by Theorem 4.3)J -

%

56

U œ 8 Ê RÐ!ß "Ñß Ð&Þ%&ÑÐ Ñ 7

+% %

%

‡MM ‡MM‡MM

Ä!

9 % "

%È where

7 œ .Ð,I J ÐIÑÑß"

/ "‡MM

!

_

I( "-

9 %%

%Ð Ñ œ Þ Ð&Þ%'Ñ

l lÊ ln

By (5.33),

U œ 8 U Þ Ð&Þ%(Ñ, 7

, Ð Ñ

+% % %‡ ‡M ‡MM

‡MM

%%" % "9 %

"% % » » Èln

Hence by (5.45) and (5.43),

% %%" %

" » » 8 Ê / Þ Ð&Þ%)Ñ, 7

,%-

%ln

‡MM

Ä!

/ B,"

Let

8 Ð Ñ ´ 8 8 Þ% %%- -

As in (4.33),

Ê % -

- %-

@ Ð Ñ8 Ð Ñ Ê RÐ!ß "Ñß Ð&Þ%*Ñ

7 Ð Ñ

Ä!%

%

with and defined as before. Thus,7 Ð Ñ @ Ð Ñ - -

% % - %% "

% %" % » » 8 œ 8 Ð Ñ 7Ð Ñ , 7

,% %

- ln‡MM

8 Ð Ñ Þ Ð&Þ&!Ñ7 , 7Ð Ñ

,% - %

% %" %

" % » » ‡MM

ln

By calculation,

7 7Ð Ñ œ 7 Ð Ñà Ð&Þ&"Ñ,

,‡MM

%"

"%% -» »ln

hence the second term in (5.50) converges to . By (5.48) and (5.50),$!

57

V œ 8 Ê / Þ Ð&Þ&#Ñ" 7Ð Ñ

Ð Ñ% %

%5 % %

% Ä!

/B,"

II. We eliminate here the restriction on . As in Theorem 4.3, let haveJ Ð!Ñ − Www !"ß. "

spectral function such that is monotone increasing inJ Ð † Ñ Ð3Ñ J ÐIÑ J ÐIÑ Iß Ð33Ñ! !

ÐJ Ñ Ð!Ñ œ , Ð333Ñ ÐJ Ñ Ð!Ñ œ + ! J Ð † Ñ! w ! ww ‡, and . Let be defined by (4.37). Weemploy the definitions between (4.37) and (4.40). Equation (4.38) holds here also, andyields

J ÐBÑ Ÿ J ÐBÑ ÐB − ß !Ñß Ð&Þ&$ÑV V% %‡ ‘ %

where

V œ 8 ß V œ 8 Þ7Ð Ñ 7Ð Ñ

% % % %‡ ‡% %

% %

% % We use arguments like those in (5.26-27) to conclude

I œ SÐ Ñß I œ SÐ ÑÞ4 4

, ,% %4

# ‡ #4

% %% %

Thus, as in (5.27-29),

[ ´ 8 Ê ß [ ´ 8 Ê ß Ð&Þ&%Ñ% % % %% %4 4 Ä! Ä!

4 ‡ ‡ 4

4 4 v v v v% W % W

where and are defined as in (5.22). As in (5.34),8 8% %4 4

‡ v v

lim inf lim inf% %Ä! Ä!

V VJ ÐBÑ Ÿ J ÐBÑß Ð&Þ&&Ñ% %‡ 4

where

V œ 8 ß 8 œ 8 8 Þ7Ð Ñ

% % % % %4 4 4

4%%

% In (5.55), we have used Lemma 5.1, and

J ÐBÑ Ÿ J ÐBÑ Ð4 − ß 5 œ 4 "Ñß Ð&Þ&'ÑV V% %

‡ß4 5 v v

where

V œ 8 8 " 7Ð Ñ

Ð Ñ% %%

4

4

v

v5 % %

% and is defined similarly. By Lemma 5.1 and (5.55)V%

‡ß4 v

58

lim inf lim inf lim inf% % %Ä! Ä! Ä!

V [ [ VVJ ‡J ÐBÑ Ÿ J ‡J ÐBÑ œ J ÐBÑ Ð&Þ&(Ñ% % % %%‡

4 44

where

[ ´ 8 Ð !ß 4 − ÑÞ% %4 4% %

Define and as in (4.25), with respect to the decomposition8 8% %‡M ‡MM

Y Y Y% % %‡ ‡M ‡MMœ ß

where corresponds to the discontinuities of and to those ofY Y% %%‡M ‡MMJ ÐIÑ’ “!

’ “JÐIÑJ ÐIÑ!

% Þ By Theorem 4.3,

9 % "

%

Ð Ñ 7

+8 Ê RÐ!ß "Ñß Ð&Þ&)ÑÈ %

%

‡MM‡MM

Ä!

with as in (4.42). The measure satisfies the hypotheses of Part I, so that7‡MM !.

%%

% 8 Ê / ß Ð&Þ&*Ñ7 Ð Ñ

%%

‡M /‡M

Ä!

B,"

where

7 Ð Ñ œ 7 ß 7 œ .ÐJ ÐIÑ ,IÑÞ Ð&Þ'!Ñ, "

/ "‡M ‡M ‡M !,

!

_

I%

"

¹ Š ‹¹ (ln "%

"

Combining (5.58) and (5.59) yields

V œ 8 8 7 Ð Ñ Þ Ð&Þ'"Ñ7 Ð Ñ 7Ð Ñ

% % %‡ ‡M ‡MM ‡M

‡M

% % %% %

% % We have

7Ð Ñ 7 Ð Ñ œ 7% %‡M ‡MM

so that by (5.58), the second term on the right of (5.61) converges to , and$!

V Ê / Þ Ð&Þ'#Ñ%%

‡ /

Ä!

B,"

By Lemma 5.1, (5.57), (5.53), and (5.62),

/ ‡J œ J ‡J ÐBÑ Ÿ J ÐBÑ Ÿ J ÐBÑ Ð&Þ'$Ñ/

Ä! Ä!V [ V V

Ä!

B,

4,

‡4

"

" % % % %W % % %Š ‹ lim inf lim inf lim sup

Ÿ J ÐBÑ œ / Þlim%Ä!

V/

%

"

‡

B,

The limit yields4 Ä _

59

J ÐBÑ J ÐBÑ Ð&Þ'%ÑW $Š ‹"4

,!

Ò4Ä_

so that

V Ê / Þ%/

B,"

III. We finally consider general for which is not identically 1. We have- 1%

- œ 1Ð!Ñ8 Ð- 1Ð!Ñ8 ÑÞ% % % %

By Theorem 3.1,

9 %%

Ð Ñ - 1Ð!Ñ8 Ê RÐ!ß @ Ñß Ð&Þ'&Ñ7 % %

%

"

Ä!"

with

7 œ . ß @ œ . Þ1ÐIÑ 1Ð!Ñ Ð1ÐIÑ 1Ð!ÑÑ /

/ " Ð/ "Ñ" "

! !

_ _

I I #

# I( (" "

"

. .

Hence

% % %% %

% % % - œ Ð- 1Ð!Ñ8 Ñ 1Ð!Ñ8 Ê / ß7Ð Ñ 7 7Ð Ñ 7

% % % %%

" "

Ä!

/

B1Ð!Ñ,"

Ð&Þ''Ñ

completing the proof. è

60

Chapter 6

Physical Applications

In this chapter we examine particle number and energy distributions in certaincanonical ensembles. The means of energy distributions provide generalized Plancklaws; the classical "blackbody curve" will be shown to apply in several situations.Normality of observables is however not guaranteed, and the extreme value distributionarises in one-dimensional systems.

To simplify the discussion, we consider only particles with chemical potential 0(whose energy of creation has infimum 0), e.g., photons and neutrinos. We study these inMinkowski space and Einstein space (a spatial compactification of Minkowski space; seeSegal [Se2]), two versions of reference space. The canonical relativistic single particleHamiltonian in its scalar approximation will be used in both cases.

We study Minkowski space ensembles in arbitrary dimension. The Hamiltonian iscontinuous, and a net of discrete approximations based on localization in space will beintroduced. Einstein space will be considered in its two- as well as its (physical) four-dimensional version. The approach in these cases can be used to establish photonnumber and energy distributions in a large class of Riemannian geometries, once therelevant wave equations are solved.

§6.1 The Spectral Measure: An Example in Schrödinger TheoryWe begin with remarks about obtaining the spectral measure for an operator with. E

continuous spectrum, acting in Minkowski space. An illustrative example of such aninfinite volume limit is in [Si].

Let be an operator on . Let denote the characteristic function ofL P Ð Ñ# 8V‘ ;

F œ ÖB À lBl Ÿ V× 1 − GV V -_ and be its volume. Suppose that for all 7

- 7 ;Ð1Ñ ´ Ð 1ÐLÑÑlimVÄ_

V"

VTr

exists. This defines a positive linear functional on , and there is a Borel measure G .-_ .

defined by

- - . -Ð1Ñ œ 1Ð Ñ. Ð ÑÞ(This measure is the . It is the spectral measure associated with an infinitedensity of statesvolume limit, and the term is reserved for situations involving Schrödinger operators.

If is a Schrödinger operator (relevant in our context for the statistical mechanics ofLnon-relativistic particles), we give a criterion for existence of a density of states [Si].Assume the dimension of space is greater than 2, and define the space of potentials8

61

O œ Z À Ä lB Cl lZ ÐCÑl.C œ ! Þ88 8#

Ä! B lBClŸ – — Ÿ(‘ ‘ : lim sup

! !

The spaces are more relevant to properties of Schrödinger operators thanO Z8 ?are or local spaces. We haveP P: :

Theorem 6.1: : Let act on , with . The density of(see [Si]) L œ Z ÐBÑ Z − O? ‘88

states exists if and only if

_ 7 ;8 VVÄ_

V" >LÐ>Ñ œ Ð / Ñlim Tr

exists.This density of states is the physically appropriate spectral measure for statistical

mechanics.

§6.2 The Spectral Approximation TheoremLet , and be a spectral measure with spectral function ; let 1 − K − W JÐ † Ñ E" " %. 1ß

w

and be the -discrete operator and occupation number r.v.'s corresponding to . Let8%4w % .

ÖE × 8% % % be a net of discrete operators with correponding occupation numbers , and4

- œ 1ÐI Ñ8 ß% % %"4œ"

_

4 4 (6.1)

with defined similarly. We now ask, How similar must the spectra of and be in- E E% %%w w

order that and coincide asymptotically in law?- -% %w

Let be the spectrum of the cardinality of , andY Y% % % % %œ ÖI × E ß R Ð+ß ,Ñ Ò+ß ,Ó4 4œ"_

- Ð+ß ,Ñ ´ 1ÐI Ñ ÐI Ñ8% % % %"5œ"

_

4 4 4;

be the truncation of with the characteristic function of . We make similar- ÐIÑ Ò+ß ,Ó%, ;definitions for , and , with respect to . We will assume that there are positiveY% % % %

w w w wß R - Eextended real functions and such thatI Ð Ñ I Ð Ñ" #% %

Ð3Ñ !i %

iÐ- Ð!ßI Ð ÑÑÑ

Ð- Ñ%

%

w"w Ò

% Ä !

Ð33Ñ "i %

iÐ- Ð!ßI Ð ÑÑÑ

Ð- Ñ%

%

w#w Ò

% Ä !

Ð333Ñ !supI Ð ÑŸIŸI Ð Ñ" #% %

¹ ¹R Ð!ßIÑR Ð!ßIÑ%

%w Ò

% Ä !

Ð3@Ñ R Ð!ßIÑ Ÿ Q R Ð!ßQ IÑ Ð! Ÿ I Ÿ I Ð ÑÑ% %" # "w %

62

for some , independent of .Q ß Q !" # % We omit the proof of

Theorem 6.2: Under these assumptions, if then and have the sameJ Ð!Ñ œ ! - -w w% %

asymptotic law i.e.,

V œ - Ð'Þ#Ñ" 7

Ð- Ñ%

%%È i %w

w

and

V œ - " 7Ð Ñ

Ð- Ñ%

%%

w È i

%

%

have identical (normal) asymptotic distributions, where

7 œ . ß1ÐIÑ

/ "w

!

_

I( ". Ð'Þ$Ñ

and

7Ð Ñ 7 œ 9Ð"Ñ Ð Ä !ÑÞ% %w

Corollary 6.2.1: Conditions and may be replaced by the generally strongerÐ3Ñ Ð33Ñones

Ð3 Ñ Ð- Ð!ßI Ð ÑÑÑ Ä !w w"%i %%

Ð33 Ñ Ð- ÐI Ð Ñß _ÑÑ Ä !Þw w#%i %%

§6.3 Observable Distributions in Minkowski SpaceWe now construct the canonical single particle Hilbert space for -dimensional8 "

Minkowski space; the constructions on other spaces are made similarly. All particles willbe treated in scalar approximations, so fields will be approximated by scalar fields, andspins by spin 0. Distributions will be studied in the frame in which expected angular andlinear momentum vanishes.

We let dimensions correspond to position8

ÐB ß B ßá ß B Ñ œ" # 8 x

and one to time . Total space-time coordinates areB œ >!

ÐB ß B ßá ß B Ñ œ Bà! " 8 ˜

63

the speed of light is 1 here.The field of a single Lorentz and translation invariant non-self-interacting9Ð † Ñ

scalar particle satisfies

?9 9 9ÐBÑ ÐBÑ œ 7 ÐBÑß Ð'Þ%ј ˜ ˜>>#

where is mass, subscripts denote differentiation, and7

? œ á Þ` ` `

`B `B `B

# # #

" ## #

8#

Since by assumption, we have .˜ ˜7 œ ! ÐBÑ œ ÐBÑ?9 9>>

The single particle space consisting of solutions of (6.4) is formulated most easily[in terms of Fourier transforms. We have if is real-valued and9 [ 9Ð † Ñ − Ð † Ñ

9 1ÐBÑ œ Ð# Ñ / JÐ5Ñ.Bß Ð'Þ&ј ˜˜ 35†B8#

8"(‘

˜ ˜

where

5 œ Ð5 ß 5 ßá ß 5 Ñß .5 œ .5 .5 á.5 Ð'Þ'ј ˜! " 8 ! " 8

and

5 † B œ 5 B 5 B Þ˜ ˜ ! ! 3 3

3œ"

8"Above, is a distributionJÐ † Ñ

JÐ5Ñ.5 œ Ö Ð5 l lÑ0Ð Ñ Ð5 l lÑ0Ð ×l .˜ ˜ ) | ,$ $! !"k k k k k k

where

0Ð Ñ− P Ð Ñà Ð'Þ(Ñ

kkÈ # 8‘

k k kœ Ð5 ßá ß 5 Ñß l l œ 5 ß . œ .5 .5 á.5 ß Ð'Þ)Ñ" 8 " # 8

3œ"

8

3# "

"#

$Ð † Ñ 0Ð † Ñ 0Ð † Ñ denotes the Dirac delta distribution, and is the complex conjugate of .The inner product of (we use and interchangeably) is0 Ð † Ñß 0 Ð † Ñ − Ð † Ñ 0Ð † Ñ" # [ 9

Ø0 ß 0 Ù œ . Þ Ð'Þ*Ñ0 Ð Ñ0 Ð Ñ

l l" #

" #(‘8

k kk

k

Since (6.5) is hyperbolic, solutions correspond to Cauchy data; we have

64

0Ð Ñ œ l l / Ð!ß Ñ. 3 / Ð!ß Ñ. Ð'Þ"!Ñ"

#Ð# Ñk k x x x x

19 98

# 8 8 Ÿ( (

‘ ‘

3 † 3>

k x k x† .

The state space thus consists of admissible Cauchy data, with time evolution dictated by(6.6). Since solutions are real, the most convenient representation of , which is9 [Ð † Ñcomplex, is as Lebesgue measurable functions on satisfying (6.7), with inner0Ð † Ñ ‘8

product (6.9). In this representation the Hamiltonian is multiplication by E œ l lÞ" `3 `> k

The operator has continuous spectrum, necessitating a discrete approximation inEthe formation of a density operator. To this end, we localize in space, replacing with‘8

the torus . Asymptotic distributions are largely independent of the compact manifold“8

used; the sphere will prove to give the same asymptotics.Let be the single particle Hamiltonian on the Torus of volume , withE% “8 Ð# Ñ1

%

8

spectrum

Y % ™% %œ 3 À Ð3 ßá ß 3 Ñ − Ð Ñ ´ ÖI × Ð'Þ""Ñ Ÿ"4œ"

8

4 4œ"# 8 _

" 8 4

"#

.

(We neglect the 0 eigenvalue, which is without physical consequence.) The particlenumber and energy r.v.'s corresponding to in its canonical ensemble (at given inverseE%

temperature) will be

8 Ð+ß ,Ñ œ 8 ß Ð+ß ,Ñ œ I 8 Þ% % % % %" "+ŸI Ÿ, +ŸI Ÿ,

4 4 4

% %4 4

?

Definition 6.3: O Ð † Ñ" denotes the modified Bessel function of the second kind, of order1.

Let be the -discrete operator corresponding to the spectral measure with spectralE%w % .

function

JÐIÑ œ ÐI   !Ñß Ð'Þ"#ÑI1

>

8# 8

8##Š ‹

which is the volume of an -dimensional sphere of radius .8 I

Lemma 6.4: Let be the number of non-zero lattice points within a sphere of radius6 ÐBÑ8

B 8 in dimensions. Then

There is a number such that +Ñ Q 6 ÐBÑ Ÿ Q B8 8 88

65

,Ñ For each 8

6 ÐBÑ

BÄ "Þ Ð'Þ"$Ñ

88##

8

>

1

Š ‹8#

Proof: Statement clearly holds for large and small, and both sides are bounded forÐ+Ñ Bintermediate values. Assertion states that the ratio of the volume of a sphere to theÐ,Ñnumber of its lattice points converges to 1. è

For brevity and convenience we define the following real functions, which arefundamentally related to asymptotic means and variances in canonical ensembles onRiemannian manifolds. They are quantities determined by "global" properties of theappropriate spectral measure , and thus arise in those less singular situations in which.asymptotics are determined by the totality of rather than its behavior near 0..

Definition 6.5: For we define8 œ !ß "ß #ßá

7 Ð+ß ,Ñ œ .Iß @ Ð+ß ,Ñ œ .IÞI I /

/ „ " Ð/ „ "Ñ8 8„ „

+ +

, ,8" 8" I

I I #( (" "

"

The Riemann zeta function is denoted by , and'Ð † Ñ

!1

>8

8##

œ Þ8

8#

Š ‹In the (important) study of total particle number and energy, we will consider@ œ @ Ð!ß_Ñ 7 œ 7 Ð!ß_Ñ8 8 8 8„ „ „ „ and , which can be calculated explicitly:

7 œ Ð8 œ #ß $ß %ßá ÑÐ8Ñ Ð8Ñ

@ œ Ð8 œ $ß %ß &ßá ÑÐ8Ñ Ð8 "Ñ

7 œÐ8Ñ Ð8ÑÐ" # Ñ à Ð8   #Ñ

Ð #Ñ à Ð8 œ "Ñ

@ œÐ8Ñ Ð8 "ÑÐ" # Ñ à Ð8  

8

8

8

8

8

"8 8

"

8

#8 8

> '

"> '

"

> ' "

"

> ' "

œÚÛÜ

ln$Ñ

#à Ð8 œ #Ñ

Ð# Ñ à Ð8 œ "Ñ

"

"

#

"

ln .

Theorem 6.6: Under symmetric statistics,

66

(1) If or if + ! 8   $ß

È % !!

%8 Ð+ß ,Ñ Ê RÐ!ß @ Ð+ß ,ÑÑ Ð'Þ"%Ñ

7 Ð+ß ,Ñ 9Ð"Ñ%

%

8 8

Ä!8 8

.

(2) If and ,+ œ !ß , ! 8 œ #

Ê % ! 1

% % "l l8 Ð!ß ,Ñ Ê R !ß Ð'Þ"&Ñ

7 Ð+ß ,Ñ 9Ð"Ñ

ln %%

8 8

Ä! #.

(3) If and ,+ œ !ß , ! 8 œ "

% "%"% "% 8 Ð!ß ,Ñ Ð" / Ñ l Ð Ñl Ê #/ O Ð#/ ÑÞ Ð'Þ"'Ñ# #

%"

%ln ln ,

Ä!"

" "B B# #

Note that (6.16) is entirely free of the global parameters of Definition 6.5.

Proof: We first show the hypotheses of Theorem 6.2 to be satisfied in cases (1) and (2).Let and correspond as before to and , andR R E E% %% %

w w

I Ð Ñ œ l l ß I Ð Ñ ´ _Þ" #% % % %È ln"%

In case (1) (if ), the lowest eigenvalue of is+ œ ! E%w

I œ SÐ Ñß%"w %

"8

while

R Ð!ßI Ð ÑÑ œ S œ S l l Þ Ð'Þ"(ÑI Ð Ñ

%w

""#

% %%

% Š ‹È ln

Hence

i % % %Ð8 Ð!ß I Ð ÑÑÑ Ÿ S l l Þ%w

" Š ‹È#8 ln (6.18)

In case (2),

i % % %Ð8 Ð!ß I Ð ÑÑÑ Ÿ S l l%w "

" Š ‹È ln . (6.19)

Equations (6.18-19) prove (6.14). Lemma 6.4 gives (6.15) and (6.16); hence the first twocases follow from Theorem 6.2.

If and has spectrum8 œ " + œ !ß E%

Y % % % % % %% œ Ö ß ß # ß # ß $ ß $ ßá×Þ Ð'Þ#!Ñ

Hence by Theorem 5.3 and the double multiplicities, the left side of (6.15) converges tothe convolution of with itself, which is on the right. / è/ B"

67

If , the distributions are explicitly8 œ 8 Ð!ß_Ñ% %

È Š ‹ Š ‹%' 1 ' 1

" > % " >8 Ê R !ß Ð'Þ#"Ñ

8x Ð8Ñ 9Ð"Ñ 8x Ð8 "Ñ%

%

8 8# #

8 88# 8## #

Ä!

if 8   $à

Ê % 1 1

% " % "l l $8 Ê R !ß

9Ð"Ñ

ln %%

$

# #Ä!

if and8 œ #à

% "%"% Š ‹8 l Ð Ñl Ê #/ O #/#

%%

lnÄ!

"

B B# #" "

if .8 œ "For , let represent the total energy of bosons in the above! Ÿ + , _ Ð+ß ,Ñ?%

ensemble whose energies lie between and .+ ,

Theorem 6.7: In a boson ensemble in -dimensional Minkowski space, the8 "asymptotic energy distribution is given by

È % ? !!

%%

%Ð+ß ,Ñ Ê RÐ!ß @ Ð+ß ,ÑÑÞ Ð'Þ##Ñ

7 Ð+ß ,Ñ 9Ð"Ñ8 8"

Ä!8 8#

Proof: The hypotheses of Theorem 6.2 can be verified as above for ; the result?%Ð+ß ,Ñthen follows from Theorem 3.14 and (6.12). è

Definition 6.8: If denotes a symmetric or antisymmetric energy r.v.,?%Ð+ß ,Ñ

WX ?

X ?%

%

%Ð+ß ,Ñ œ Ð'Þ#$Ñ

Ð Ð+ß ,ÑÑ

Ð Ð!ß_ÑÑ

is the corresponding to . If the limitenergy distribution function ?%Ð+ß ,Ñ

WX ?

X ?Ð+ß ,Ñ œ ß Ð'Þ#%Ñ

Ð Ð+ß ,ÑÑ

Ð Ð!ß_ÑÑlim%

%

%Ä!

exists, it is the of the net. If is absolutelyasymptotic energy distribution WÐ!ßIÑcontinuous,

68

.ÐIÑ œ Ð!ßIÑ.

.IW

is the asymptotic energy density of the net.

Theorem 6.7 gives the asymptotics of if , then% ? ? ?Ð Ð+ß ,ÑÑ À œ Ð!ß_Ñ% % %

È Š ‹% ?' 1

%" >%

% Ê RÐ!ß @Ñß

88x Ð8 "Ñ 9Ð"Ñ8#

8" 8##

Ä!

where

@ œ Þ Ð'Þ#&Ñ8Ð8 "Ñx Ð8 "Ñ' 1

" >

8#

8# 8##Š ‹

Corollary 6.9: The asymptotic energy density of the Minkowski -space boson8 "canonical ensemble is

. ÐIÑ œ ÐI   !ÑÞ Ð'Þ#'ÑI

Ð/ "Ñ8x Ð8 "ÑF

8 8"

I

"

'"

We now find the asymptotic fermion distributions corresponding to .E%

Theorem 6.10: Let be the fermion number in the Minkowski space canonical8 Ð+ß ,Ñ%

ensemble. Then

È % !!

%8 Ð+ß ,Ñ Ê RÐ!ß @ Ð+ß ,ÑÑÞ Ð'Þ#(Ñ

7 Ð+ß ,Ñ 9Ð"Ñ%

%

8 8

Ä!8 8

Similarly, for fermion energy,

È % ? !!

%%

%Ð+ß ,Ñ Ê RÐ!ß @ Ð+ß ,ÑÑÞ Ð'Þ#)Ñ

7 Ð+ß ,Ñ 9Ð"Ñ8 8"

Ä!8 8#

Hence if ? ?% %œ Ð!ß_Ñ

È Š ‹% ?' 1

%" >%

% Ê RÐ!ß @Ñß Ð'Þ#*Ñ

8x8 Ð8 "Ñ Ð" # Ñ 9Ð"Ñ8# 8

8" 8##

Ä!

with

69

@ œ Þ Ð'Þ$!Ñ8Ð8 "Ñx Ð8 "Ñ Ð" # Ñ' 1

" >

8# 8

8# 8##Š ‹

Corollary 6.11: The fermion ensemble in Minkowski -space has asymptotic energy8 "density

. ÐIÑ œ Þ Ð'Þ$"ÑI

Ð/ "Ñ8x Ð8 "ÑÐ" # ÑJ

8 8"

I 8

"

'"

§6.4 Distributions in Einstein SpaceWe now consider distributions in two- and four-dimensional Einstein space,

Y œ W ‚ Y œ W ‚ E# " % $‘ ‘ and . In four dimensions the single particle Hamiltonian %

is the square root of a constant perturbation of the Laplace-Beltrami operator for whichthe wave equation satisfies Huygens' principle.

We first consider Einstein 2-space, with

% œ ß Ð'Þ$#Ñ"

V

where is the radius of the spatial portion . Clearly has spectrum (6.20), and theV W E"%

V Ä _ asymptotics follow directly from Theorems 6.6-10. Since the discrete pre-asymptotic operator is of more direct interest here, we willE%

analyze associated distributions more carefully. We define two combinatorial functions.

Definition 6.12: If and , then! Ÿ + , Ÿ _ß 6 − ™

: Ð+ß ,Ñ ´ Ð8Ð3Ñ "Ñ Ð'Þ$$Ñ6Ð#Ñ

8ІÑ−T Ð+ß,Ñ +Ÿ3Ÿ,

" $6

where

T Ð+ß ,Ñ œ 8Ð † Ñ − À 38Ð3Ñ œ 6 ß Ð'Þ$%Ñ6 Ò+ß,Ó

+Ÿ3Ÿ, Ÿ"a

and is the set of all functions . Define as thea ™ ™Ò+ß,Ó

6Ð#Ñ

8Ð † Ñ À Ò+ß ,Ó Ä ; Ð+ß ,Ñ

number of ways of expressing as a sum of integers in the interval , no single6 − Ò+ß ,Ó™

integer being used more than .twice

We present the following without proof.

Theorem 6.13: If and is the boson energy in Einstein 2-space! Ÿ + , Ÿ _ Ð+ß ,Ñ?%

at inverse temperature , then is concentrated on" ? ! Ð+ß ,Ñ%

70

%™ % ™ œ Ö 4 À 4 − ×ß Ð'Þ$&Ñ

and

c ? % ™Ð Ð+ß ,Ñ œ 6Ñ œ Ð6 − Ñß Ð'Þ$'Ñ: Ð+ß ,Ñ/

O%

%"6Ð#Ñ 6

F

where

O œ Ð" / Ñ Þ Ð'Þ$(ÑF

4œ"

_ 4 #$ "%

If represents fermion energy, then is replaced by , and by?% : ; O6 6Ð#Ñ Ð#Ñ

F

O œ Ð" / Ñ Þ Ð'Þ$)ÑJ

4œ"

_ 4 #$ "%

The limits of the two-dimensional distributions are specializations of the V Ä _ 8 œ "cases in §6.1.

Proposition 6.14: As , the asymptotic energy densities of bosons and"% œ V Ä _

fermions in Einstein 2-space are

. ÐIÑ œ à . ÐIÑ œ Þ'I "#I

Ð/ "Ñ Ð/ "ÑF J

# #

I # I ## #

" "

1 1" "

We now consider distributions in Einstein 4-space . The Hamiltonian acts asW ‚ E$ ‘ %" `3 `7 (where is time) on the Hilbert space of solutions of the invariant non-self-7

interacting scalar wave equation. It has spectrum having5 % %ÐE Ñ œ Ö8 À 8 − ×ß 8%

multiplicity .8#

Proposition 6.15: If then and are trace class." ! / / . ÐE Ñ . ÐE Ñ" > " >F J% %

Proof: By Proposition 1.4 it suffices to show is trace class; we have/ E" %

tr / œ 4 / œ / _Þ è Ð'Þ$*Ñ/ "

Ð/ "Ñ E # 4

4œ"

_

$" "% "%

"%

"%% "

Note that

71

O œ / œ Ð" / Ñ Ð'Þ%!Ñ

O œ / œ Ð" / Ñ Þ

F . ÐE Ñ 4 4

4œ"

_

J . ÐE Ñ 4 4

4œ"

_

tr

tr

" > "%

" > "%

F#

J#

%

%

$$

For completeness we derive the exact distribution of energy in terms of the followinginteger-valued combinatorial functions.

Definition 6.16: If and , let! Ÿ + , Ÿ _ 6 − ™

: Ð+ß ,Ñ œ ß Ð'Þ%"Ñ8Ð4Ñ 4 "

4 "6Ð4 Ñ

8ІÑ−V Ð+ß,Ñ +Ÿ4Ÿ,

#

#

#

6

" $ where

V Ð+ß ,Ñ œ 8Ð † Ñ − À 8Ð3Ñ œ 6 ß6 Ò+ß,Ó

+Ÿ3Ÿ, Ÿ"a

with as in Definition 6.12, and denoting choose Let be theaÒ+ß,Ó 6Ð4 ÑŠ ‹-

.- .Þ ; Ð+ß ,Ñ

#

number of ways (without regard to order) of expressing as a sum of integers in6 − ™

Ò+ß ,Ó 4 4, each integer being used no more than times.#

Theorem 6.17: The Bose energy in Einstein 4-space at inverse temperature?%Ð+ß ,Ñ" %™ ! ß, is concentrated on and

c ? % ™Ð Ð+ß ,Ñ œ 6Ñ œ Ð6 − Ñà Ð'Þ%#Ñ: Ð+ß ,Ñ/

O%

"%6Ð4 Ñ 6

F

#

3 : ; O O Þn the Fermi case is replaced by , and by 6 6Ð4 Ñ Ð4 Ñ

F J

# #

We now consider asymptotics of Einstein-space distributions. Denote by the -E%w %

discrete operator corresponding to the spectral measure for which . Let. JÐIÑ œ I'

$

R Ð!ßIÑ E Ñ Ð!ßIÑ R Ð!ßIÑ% % % be the cardinality of ( , with the analogous definition for 5 w

relative to .E%w

Lemma 6.18: If and , then % ! I   ! R Ð!ßIÑ Ÿ R Ð!ß #IÑÞ% %$w

72

Proof: We have

R Ð!ß #IÑ œ ß%I

$%$w

$

#– —%

while

R Ð!ßIÑ œ 4 œ " # " Þ Ð'Þ%$Ñ" I I I

'% " – —– — – —

4œ"

#

’ “I%

% % %

We have

– — – — – — – —I I I %IŸ Ÿ Ÿ Ð !ß I   !Ñß Ð'Þ%%Ñ

$% % % %%

# $$

$

and equations (6.42-43) imply the result. è

Lemma 6.19: Let be the boson number corresponding to in Einstein 4-space.8 Ð!ßIÑ E% %

Then

% i % Ò$

Ä!Ð8 Ð!ß ÑÑ ! Ð'Þ%&Ñ%

%È

and

supI 

w Ä!È%

%

%%» »R Ð!ßIÑ

R Ð!ßIÑ " !Þ Ð'Þ%'Ñ

$

Ò

Proof: We have

i %"%

"%

% %

"

Ð8 Ð!ß ÑÑ œ Ÿ / Ð'Þ%(Ñ/ "

Ð/ "Ñ Ð Ð'4Ñ Ñ

Ÿ B .B/

Ð Ñ

œ ß$

%

"%

"%

" %

" %

"%

$

$

' $ "$

" "$

"'

$

"

#$

&#

w

4œ" 4œ"

Ð'4Ñ

Ð'4Ñ # #

#!

#

È " "(

’ “ È “

È

È È

È

%

% % %

% %

from which (6.45) follows. Equation (6.46) is implied by

R Ð!ßIÑ œ Ð'Þ%)ÑI

'%$w

$

$– —%

and (6.43).

73

Lemma 6.20: If , then! Ÿ + , Ÿ _

R Ð+ß ,Ñ

R Ð+ß ,Ñ"ß

%

%%

$w Ä!

Ò

and

% i Ò$ w

Ä_Ð8 ÐIß_ÑÑ ! Ð'Þ%*Ñ% %

uniformly in .%

Proof: The first assertion follows from (6.46). The second follows from the existence offixed such that%‡ ‡ß I !

R ÐIßI Ñ Ÿ #R ß I Ð Ÿ ß I Ÿ I I _ÑßI

#% %

w w ‡ ‡" "$ % %

and the fact that

% i Ò$ w

IÄ_Ð8 ÐIß_ÑÑ ! Ð'Þ&!Ñ%$

uniformly in . % è

Theorem 6.21: The boson number in Einstein 4-space satisfies

È % Ò%

%$

$ Ä!8 Ð+ß ,Ñ RÐ!ß @Ñß

7Ð Ñ%

%

where

7Ð Ñ œ .I 9Ð"Ñ Ð Ä !Ñ Ð'Þ&"Ñ" I

# / "% %(

+

, #

I"

+8.

@ œ .IÞ" I /

# Ð/ "Ñ(+

, # I

I #

"

"

If represents fermion number, each “ ” is replaced by “ ”.8 %

Proof: This follows in the Bose case from Lemmas 6.18-20, Theorem 6.1 and itsCorollary, and Theorem 3.14. è

74

Theorem 6.22: The Einstein 4-space asymptotic boson and fermion number and energydensities are identical to those in Minkowski 4-space. Precisely, equations (6.14), (6.22),and (6.27-28) still hold.

Corollary 6.23: The asymptotic energy densities of Bose and Fermi ensembles in Einstein4-space are

. ÐIÑ œ"&I

Ð/ "ÑF

$ %

I %%

"

1"

. ÐIÑ œ Þ"#!I

(Ð/ "ÑJ

$ %

I %%

"

1"

§6.5 Physical DiscussionThe requirements of localization in physical (as well as momentum) space result in

the approximation in §6.2 of the Hamiltonian for massless scalar particles in Minkowskispace. There seems to be no direct way of discretizing without reference to .. ÐEÑ E>The r.v. is interpreted as boson number in an -dimensional torus of volume8 8%

Z œ 8   #Ð# Ñ 7Ð# Ñ

1% 1

!88 8

8. If the expected number per unit volume is asymptotically ; thesame density occurs for Einstein 4-space (Theorem 6.21). If the mean density of8 œ "

photons in the energy interval is asymptotically , and thus large (andÒ!ß ,Ó Ð Ä !Ñl Ð Ñlln "%1" %

V-dependent) in Einstein space and infinite in Minkowski space. Since asymptoticdensity is independent of , the divergence clearly arises from bosons with effectively,vanishing energy; this is an effect of Bose condensation, in which large numbers ofparticles appear in the lowest energy levels. The corresponding spatial energy density is,however, finite according to Theorem 6.7, which gives mean density

H œ .IÞ8 I

# / "8

8 8##

!

I

I1 >

8#

#

Š ‹ ( "

This divergent density of low-energy bosons with finite corresponding energy density hasan analog in the "infrared catastrophe" of quantum electrodynamics (see [BD], §17.10),in which an infinite number of "soft photons" with finite total energy is emitted by anelectrical current. The extreme density is correlated with the lack of normality in photonnumbers.

The energy distribution of bosons in a Minkowski -space canonical ensemble in8 "(6.25) is the “Planck law” for such a system. An observer of scalar photons measuresthe proportion of photon energy in the frequency interval to beÒ ß Ó/ / ?/'2

2Ð Ñ

F/

/ ?/. ÐIÑ.I 2, with Planck's constant. Analogously, (6.31) gives the

corresponding law for fermions; in an ensemble consisting of neutrinos, the proportion oftotal neutrino energy observed in is thus . Note that theÒIßI IÓ . ÐIÑ.I? '

I

I I

J

?

75

Planck laws for Einstein 4-space coincide asymptotically with those in Minkowski space;this obviously also holds in one dimension. This indicates that the cosmic backgroundradiation expected in an approximately steady-state model of the universe is largelyindependent of the underlying manifold. The specific correspondences in this chapter area consequence of the physical identity of Minkowski space and the “ limit” ofV Ä _Einstein space of radius .V

76

Chapter 7

The Lebesgue Integral

In this chapter we introduce a natural and useful generalization of the notions inChapter 3. Lebesgue integration of r.v.-valued functions on a measure space is themaximal completion of Riemann integration. The step from Riemann to Lebesgueintegration shifts the focus from the domain to the range of the integrated function;indeed, the ordinary Lebesgue integral is a Riemann integral of the identity function onthe range space with respect to the domain-induced measure; this viewpoint will be usedhere.

The present integration theory is in fact interpretable as a formal extension of thetheory of semi-stable stochastic processes (see [La, BDK]), with an abstract measurespace replacing time. The r.v.-valued function being integrated yields an r.v.-valued\measure defined by integral of over ; this measure is clearly a` `ÐEÑ œ \ Egeneralized stochastic process. Indeed, in the notation of this chapter, if is aÖ\Ð>Ñ×>−‘

collection of independent standard normal r.v.'s, then is simply Brownian'!

>"Î#\Ð>ÑÐ.>Ñ

motion.The advantage of the present approach to that of standard integrals of distribution-

valued random functions (e.g., “white noise” ) [Che,R1,V] is that it does not require theexistence of a metric (or smooth volume element) on the underlying measure space.Precisely, the present integration theory would, on a Riemannian manifold, be equivalentto (linear) integration of an r.v.-valued distribution with covariance

X - - i - $ - - - AÐ\Ð Ñ\Ð ÑÑ œ Ð\Ð ÑÑ Ð ß Ñ Ð − Ñ Þ" # " " # 3

(Here, denotes the point mass at in the variable ) However, an abstract measure$ - -# "ß Þspace generally has no such object.

A novel aspect of the Lebesgue integral is the use of a non-linear volume element9 .Ð. Ñ. This may seem rather ad hoc; but in Chapter 8 the Lebesgue integral is shown tobe isomorphic to a linear integral over functions with range in a space of logarithms ofcharacteristic functions.

Lebesgue integration is the natural environment for detailed study of integrals ofindependent random variables; however, aside from proof of associated fundamentalswhich will comprise most of this and the next chapter, the approach here will berelatively goal-and-applications oriented. Further on, the measure space willÐ ß ß ÑA U .be the spectrum of an abelian von Neumann algebra of quantum observables.

The material in the next three chapters will be largely independent of previousmaterial, and the probabilistic content stands on its own. The proofs may be omitted on afirst reading.

77

§7.1 Definitions and Probabilistic BackgroundWe will often deal with normalized integrals (sums) of random variables with infinite

variance; the resulting limits will depend strongly on the tails of the integrateddistributions. We recall the bare essentials of general limit theorems for sums ofindependent random variables; see [GK] for details.

In order to formulate the most general central limit results, we define the Lévy-Khinchin transform of an infinitely divisible distribution. Recall every such distribution/ F < has characteristic function , where is continuous and vanishes at Theœ / > œ !Þ<

book of Loève [Lo] has a proof of

Theorem 7.1: The distribution is infinitely divisible if and only if , where is/ F <œ /<

given (uniquely) by

< #Ð>Ñ œ 3 > / " .KÐBÑß Ð(Þ"Ñ3>B " B

" B B( _

_3>B

# #

#

with and is a non-negative multiple of a d.f.# ‘− K

The pair is the of . Note this is additive, in that ifÐ ßKÑ# /Lévy-Khinchin transform/ # / / # #3 3 3 " # " # " # has transform , then has transform . It canÐ ßK Ñ Ð3 œ "ß #Ñ ‡ Ð ß K K Ñbe shown that the transform is continuous from the space of distributions in the/topology of weak convergence, to the pairs in the topology of crossed with theÐ ßKÑ# ‘"

topology of weak convergence.Let be a double array of independent randomÖ\ ×ß " Ÿ 8 _ß " Ÿ 5 Ÿ 585 8

variables; is allowed.5 œ _8

Definition 7.2: The variables are if for every \ !ß85 infinitesimal %

sup"Ÿ5Ÿ5

858

T Ðl\ l   Ñ !Þ% Ò8 Ä _

For any monotone functions of bounded variation, we writeK ÐBÑß KÐBÑ8

K ÐBÑ Ê KÐBÑ8" if the same is true of the corresponding measures on .‘

Lemma 7.3.1: Let be a sequence of independent r.v.'s. In order that Ö\ × \8 88œ"

_!converge weakly and order-independently, it is necessary and sufficient that

" " 5œ" 5œ"

_ _#8

8 8# #

8X X

\ \

" \ " \ß Ð(Þ#Ñ

converge absolutely.

Proof: By the three-series theorem, order-independent convergence above is for any7 ! equivalent to absolute convergence of the series

78

" " "( ( (8 8 8lBl  lBlŸ lBlŸ

8 8 8#

7 7 7

.J ÐBÑß B.J ÐBÑß B .J ÐBÑß Ð(Þ$Ñ

where is the d.f. of . But absolute convergence of (7.3) implies convergence of theJ \8 8

first series in (7.2). Thus proving absolute convergence of the second series reduces tothe same for

" "( ( 8 8lBlŸ lBlŸ

# #8 8

$

7 7

B B

" B " B B .J ÐBÑ œ .J ÐBÑÞ Ð(Þ%Ñ

Convergence of the latter follows from that of the first series in (7.2).

Conversely, if (7.2) converges absolutely, so do the first and third series of (7.3),while convergence of the middle series follows from a subtraction argument, as in (7.4).è

Henceforth, will denote the (cumulative) distribution function of , and allJ ÐBÑ \85 85

infinite sums must converge order-independently to be well defined. Assume that{ form an infinitesimal array. The proof of the following theorem for finite is\ × 585 85œ"

58

in Loève [Lo]; we extend it to general .58

Theorem 7.3: Let . In order that converge weakly to a distribution ,W œ \ W W8 85 85

!i is necessary and sufficient that>

K Ê Kß Ä Þ Ð(Þ&Ñ8 8# #

Here

#8

8 8

œ + .J ÐCÑß K ÐBÑ œ .J ÐCÑß Ð(Þ'ÑC C

" C " C" "( (5œ" 5œ"

5 5

85 85 8 85_ _

_ B

# #

#

and

+ œ C.J ÐCÑß J ÐBÑ œ J ÐB + Ñß85 85 85 85 85lClŸ

(7

with any fixed constant. The pair is the Lévy-Khinchin transform of .7 # ! Ð ßKÑ W

Proof: This follows using approximation by finite sums. For example, if convergeW8

weakly to , the same is true of , where is finite but sufficiently large;W W œ \ 58 8‡ ‡

5œ"

5

85!8‡

using the result for and letting become infinite proves (7.5). The only difficulty liesW 58 8‡ ‡

in proving the sum defining converges (order-independently) if and only if (7.6) does.W8

To this end, note that if exists, the three-series theorem implies converges.W l+ l8 855œ"

_!

79

Letting ,;ÐCÑ œ C"C#

" "( (" (

5œ" 5œ"

_ _

_ _

_ _

85 85 85

5œ"

_

_

_

85 85 85w

; ;

; ;

ÐCÑ .J ÐCÑ œ ÐC + Ñ.J ÐCÑ Ð(Þ(Ñ

œ Ð ÐCÑ + ÐC ÑÑ.J ÐCÑ

where is determined according to the mean value theorem. ByC + Ÿ C Ÿ C85 85

infinitesimality of the , this sum converges if and only if \ ÐCÑ.J ÐCÑ85 855œ"

_

__! ' ;

converges. This (again three-series) occurs if and only if converges! '5œ"

_

8577;ÐCÑ.J ÐCÑ

for some . The latter follows from the Lemma. The existence of the limit 7 ! KÐBÑfollows similarly.

Conversely, assume the limits (7.6) exist. Then the sum

" (5œ"

_

85 85

+ C.J ÐCÑ7

7

converges for any , since is dominated by for . Therefore we7 ; 7 ! C ÐCÑ lCl ŸC"C

#

# have convergence of

" " "(" ( (

5œ" 5œ" 5œ"

_ _ _

85 85 85 85 85 85 85 +

+

5œ"

_ + +

85

+ ÐC + Ñ.J ÐCÑ œ + + TÐl\ + l Ñ Ð(Þ)Ñ

C.J ÐCÑ

7

7

7 7

7 7

85

85

85 85

7

.

Using infinitesimality of and finiteness of for (the latter+ T Ðl\ l   GÑ G !85 855

!follows from convergence of the second sum in (7.6)), the last two sums converge.Therefore is convergent, as is . Using (7.7) and the infinitesimality! !'

585 85_

_+ .J;

of , we conclude that converges (absolutely) as well. Similarly Ö+ × .J85 855 5

__! !' ;

'__ C

"C 85 8#

# .J _ W è. Thus, by Lemma 7.2.1, exists and is order-independent.

Henceforth let

; ; ) ;‡"B

# #

#

ÐBÑ œ ß ÐBÑ œ ß Ð Ñ œ Þ Ð(Þ*ÑBà lBl Ÿ "

à lBl   "

B B

" B " Bœ

Corollary 7.3.1: In order that converge weakly, it is necessary and sufficient thatW8

80

# )w

8Ä_5œ" 5œ"

5 5

85 85_

B

œ , ß KÐBÑ œ ÐCÑ.J ÐCÑ ˜lim lim" "(8 8

w -

exist, where

, œ ÐCÑ.J ÐCÑß J ÐCÑ œ J ÐC , Ñ Ð(Þ"!Ñ85 85 85 85 85_

_‡( ; ˜ .

The weak limit has Lévy-Khinchin transform , whereW Ð ßKÐ?ÑÑ#

# ;œ , ÐCÑ.J ÐCÑÞ ˜lim8Ä_

5œ"

5

85 85_

_" (8

Proof: If converges weakly, then for someW K ÐBÑ œ ÐCÑ.J ÐCÑ Ê K ÐBÑ8 8 855

_

B

8Ä_

‡!' )

multiple of a d.f. If are continuity points of , thenK „ K‡ ‡7

" ( (5 _ _

85 8

#

#

7 7

.J ÐCÑ œ .K ÐCÑ Ð(Þ""Ñ" C

C

converges as . Similarly, converges; and8 Ä _ .J ÐCÑ!'5

lCl  857

" ( (5 lClŸ lClŸ

$

# 85 87 7

C

" C.J ÐCÑ œ C.K ÐCÑ

converges as well, as does . Thus, by convergence of in (7.6)!'5

lCl C

"C 85 87 # .J ÐCÑ #

and the identity , we have convergence ofB œB B"B "B# #

$

" (5œ"

5

85 85lClŸ

8

+ C.J ÐCÑ ß7

with as in the Theorem. We conclude the convergence of+85

" "( ( " (

5œ" 5œ"

5 5

85 85 85 85 85 + lClŸ

+

5œ"

5

85 85 85lCl 

8 8

85

85

8

+ C.J ÐCÑ œ + ÐC + Ñ.J ÐCÑ Ð(Þ"#Ñ

œ + + .J à

7 7

7

7

the sum clearly converges to 0 as by infinitesimality of the! '5œ"

5

85 85lCl 

8

+ .J 8 Ä _7

+ + 8 Ä _85 855œ"

5

, and hence converges to a limit as .!8

81

On the other hand, since for converges weakly (equation (7.11)),B !ß J ÐBÑ! 85

( ( " "_ _

+ ‡ ‡

85 85 85

5 5

7 785

; ;ÐB + Ñ. J ÐBÑ œ ÐBÑ.J

converges, so that the sum defining converges (the sum converges by# ;w ‡

5

_85!'

7 .J

similar arguments). Let , and consider- ´ , +85 85 85

" "( (" ( (

5 5_ _

B B-

85 85 85 85

5 _ _

B- ?-

85 8585‡ w

) )

) )

ÐCÑ.J ÐC , Ñ œ ÐC - Ñ.J ÐCÑ Ð(Þ"$Ñ

œ ÐCÑ.J - ÐCÑ.J ß

85

85 85

where is between 0 and . Without loss of generality, we let ; then- - "85‡

85 7

" " "» »(5 5 5

85 85 85 85lBl 

‡l- l œ l, + l œ ÐCÑ.J ÐCÑ Ð(Þ"%Ñ7

;

remains bounded as , as does . We have 8 Ä _ l- l l ÐCÑl.J !! '85‡ w

__

85) Ò8 Ä _

uniformly in , so that the last term on the right of (7.13) vanishes as , and5 8 Ä _

" (5 _

B

85‡)ÐCÑ.J ÐCÑ Ê K Ð?Ñ œ KÐ?ÑÞ Ð(Þ"&ј

Conversely, if the series for and converge absolutely in and as , then it#w KÐ?Ñ 5 8 Ä _

follows along similar lines that converges as , for any! '5œ"

5

lCl ‡

85

8

7 ; ÐCÑ.J ÐCÑ 8 Ä _

7 ; !Þ .J 8 Ä _ß + Thus, since converges as converges as well. Using! !5œ" 5

5‡

85 85

8

by now standard arguments, it follows that converges as !'5

__

85;ÐCÑ.J ÐCÑ 8 Ä _ß

as does Similarly, if is a continuity point of ! 'Ð+ ÐCÑ.J ÐCÑÑÞ B KÐBÑß85 85__

;

then .!'5

_B

85)ÐCÑ.J ÐCÑ Ä KÐBÑ

We now prove the last statement of the Corollary. We have

lim8Ä_

5

85 85_

_"– —(+ .J œ;

82

lim8Ä_

5

85 85 85_ _

_ _‡‡"– —( (+ .J .J ß Ð(Þ"'Ñ

; ;

)) ;˜

since the measures and have the same weak limit (see (7.15)). By˜! !5 5

85 85) ).J .J

the mean value theorem

" "( ( (" (

5 5_ _ _

_ _ _‡ ‡ ‡ ‡

85 85 85 85w

85

5

85 85_

_‡ ‡w

85

; ; ;

;

.J œ .J , ÐC , ÐCÑÑ.J Ð(Þ"(Ñ

œ , Ð" ÐC , ÐCÑÑ.J Ñ Þ

˜

Since is uniformly bounded (since is),! !' '5 5

_ lCl "_ ‡ ‡w

85 85 85Ð" ÐC , ÑÑ.J .J;

and uniformly, (7.17) vanishes as Similarly, if is a continuity, ! 8 Ä _Þ85 Ò8 Ä _ 7

point of andK

;77

‡‡ÐBÑ œBà lBl Ÿ!à lBl ßœ

then uniformly. Thus (7.16) is given by!'5

__ ‡‡

85; Ò.J !8 Ä _

lim8Ä_

5

85 85 85_ _

_ _‡ ‡‡"– —( (+ Ð Ñ.J .J; ; ; ˜

œ + Ð Ñ.J .J Ð(Þ")Ñ

œ + .J .J"

C

œ , .J ß

lim

lim

lim

8Ä_5

85 85 85_ _

_ _‡ ‡‡

8Ä_5

85 85 85lCl _

_

8Ä_5

85 85_

_

"– —( ("– —( (" (

; ; ;

;

;

˜ ˜

˜ ˜

˜

7

the last equality following from

"( (5 lCl lC, l

85 857 7

" "

C C.J .J !Þ˜ ˜

85

Ò8 Ä _

Equation (7.18), together with (7.15), completes the proof. è

83

§7.2 Definition of the Lebesgue IntegralSince we will study integrals of measure-valued functions, we consider metrics on

spaces of probability distributions. Let be the set of probability Borel measures on W ‘"ßand the set of finite Borel measures. Define the Lévy metric byW 3‡ P

3 / / / / WP ‡" # " # " " #Ð ß Ñ œ Ö2 À J ÐB 2Ñ 2 Ÿ J ÐBÑ Ÿ J ÐB 2Ñ 2×ß Ð ß − Ñßinf

Ð(Þ"*Ñwhere is the d.f. of . The Lévy metric is compatible with the topology ofJ3 3/convergence in distribution on (see, e.g., [GK]). Note that can be defined for anyW 3P

pair of Borel measures on . This general definition will be used here./ / ‘" #ßWe will also require a stronger metric which emphasizes tail properties of measures.

Let : be defined for small arguments, and nonvanishing. Define (recall (7.9))9 ‘ ‘ Ä

K ÐBÑ œ ÐCÑ . Þ Ð(Þ#!Ñ" C

Ð Ñ/ $ß

_

B

$ 9 $) /(

We define

3 / / 3 ; / /$ 9 $

9 $ / $ / $ß " # ß ß " #P

_

_

Ð ß Ñ œ ÐK ßK Ñ ÐCÑ.Ð Ñ Þ" C

Ð Ñ" # » »(The measure is defined by/Ð+BÑ

/ /Ð+BÑÐGÑ œ Ð+GÑß

for a Borel set in . Since the integrand of (7.20) vanishes only at , it followsG B œ !‘that for fixed , and are equivalent. We introduce the strengthened metric$ 3 3P

ß9 $

3 / / 3 / /9 9 $$

Ð ß Ñ œ Ð ß ÑÞ Ð(Þ#"Ñ" # ß " #! "sup

This will be useful for our integration theory, in which tail behavior of probabilitydistributions will be crucial. Note that infinite distances under are not excluded; this:9is clearly an inconsequential deviation from the standard properties of a metric. We willnow require a proposition. We define for , ,0 À Ä !‘ ‘ %

Ð 0ÑÐBÑ œ Þ0ÐB Ñ 0ÐB Ñ

lipÐ Ñ

! Ÿ

%

$ %

sup » »$ $

$

If is a measure on , then is the total mass of ./ ‘ / /l l

Proposition 7.4: Let be absolutely continuous and of bounded variation, and" ‘ ‘À Ä/ / ‘ % 3 / / " "" # " # _

P w"

ß œ Ð ß Ñ m m ß m be finite Borel measures on . Let , and assume that ,¼and are finite, where denotes the derivative. Then¼ ¼lipÐ Ñ w w

"% l lÐBÑ" "

84

( ’ “¼ ¼ ¼_

_

" # _ " #w Ð Ñ w

"" / / % " " " / .Ð Ñ Ÿ m m # # l lÐBÑm l l Þlip %

Proof: If (allowing a slight abuse of notation), then/ /3 3_

B wÐBÑ œ . ÐB Ñ'» » » »( (

_ _

_ _

" # " # " #__" / / " / / / / " .Ð Ñ Ÿ l ÐBÑÐ ÑÐBÑl l Ð ÑÐBÑ. ÐBÑ

Ÿ Ð ÐB Ñ Ð ÐB Ñ ÑÑ l. ÐBÑl

Ÿ Ð # Ñ Ð ÐB Ñ ÐB ÑÑl ÐBÑl.B

œ Ð # Ñ ÐBÑÐl ÐB Ñl l ÐB

% " / % % / % % "

% " " / % / % "

% " " / " % "

¼ ¼ (¼ ¼ ¼ ¼ (¼ ¼ ¼ ¼ (

__

_

# #

_ "w w

_

_

# #

_ "w w w

_

_

# ÑlÑ.B

œ Ðm m #m m Ñ ÐBÑ. ÐBÑß

%

% " " / (_ " #w

_

_( %

where

( " % " %%ÐBÑ œ Ðl ÐC Ñl l ÐC ÑlÑ.Cß(!

Bw w

and

l ÐBÑl Ÿ m ÐBÑm Þ( % "%%lipÐ Ñ w

"

Thus

¿ ¿( (_ _

_ _

" # _ " #w _

# _" / / % " " ( / ( /.Ð Ñ Ÿ Ðm m #m m Ñ l .% %

Ÿ m m #m m #m ÐBÑm l l ß% " " " /Š ‹_ " " #w Ð Ñ wlip %

as desired. è

Remark: If is any function for which satisfies the hypotheses of; 0ÐBÑ œ ; ;)

ÐBÑ ÐBÑÐBÑ

Proposition 7.4, then a metric equivalent to obtains by replacing by in the3 ; ;9

definition. To see this, note that by Proposition 7.4

85

» » » »( (" B " BÐ Ñ.Ð Ñ œ 0 .Ð Ñ Ÿ JÐ Ð ß ÑÑß

$ 9 $ 9; ; / / ) / / 3 / /

_ _

_ _

" # " # " #9

where , and is a continuous increasing function vanishing at 0.0 œ J; ;)

Let be a -finite measure space, and be , in thatÐ ß ß Ñ \ À ÄA U . 5 A W measurable \ Ð Ñ − § Þ Ð\Ñ" b U 3 b W e for every -open Let be the range, and9

e / e . / % 3essÐ\Ñ œ Ö − Ð\Ñ À Ð\ ÐF Ð ÑÑÑ ! a !× F"% % 9, where is a -ball of radius

% eÞ Ð\Ñ Note for future reference that gives the full picture with regard to integrationesstheory, since

. - - eÖ À \Ð Ñ Â Ð\Ñ× œ !Þ Ð(Þ##Ñess

For otherwise, there would exist an uncountable number of open balls in ,ÖF ×7 7−Q Wwith

. .\ ÐF Ñ œ !ß ! \ F _" "7 7

7 .

(recall is -finite). Such a situation is equivalent (by collapsing each ) to a discreteA 5 F3

non-atomic measure space of positive finite total measure, which does not exist.By analogy with the Lebesgue integral, we initially assume is finite and is -. 3\ 9

bounded, i.e., the diameter of its range is bounded. Let be a sequence of atÖT ×! !œ"_

most countable partitions of , each with elements . Fore eess essÐ\Ñ ÖT ×ß T œ Ð\Ñ! !3 33

-W § W œ Ö Ð ß Ñ À − W×W 3 / / /, let diam sup . We assume" # 3

Ð3Ñ QÐT Ñ ´ Ð T Ñ !! !sup3

3diam Ò! Ä _

Ð33Ñ 7ÐT Ñ ´ ÐT Ñ !Þ! !sup3

3. Ò! Ä _

Hence the mesh of vanishes both in diameter and measure . For the purposesT Ð3Ñ Ð33Ñ! of this definition, as in Chapter 3, atoms of may be artificially divided into pieces : :. 4

and apportioned to various , as long as T Ð: Ñ œ Ð:ÑÞ!3 4!. .

Definition 7.5: A partition net satisfying and is .Ð3Ñ Ð33Ñ infinitesimal

Let be defined for small positive , and vanish at 0. We say that9 $ $Ð Ñ

P ( "A !

! !\Ð Ñ Ð. Ð ÑÑ ´ \Ð Ñ Ð \ ÐT ÑÑ- 9 . - - 9 .limÄ_

3 3"

exists if the right side is independent of choice of and of satisfying -! ! !3 3 3œ"_− T ÖT × Ð3Ñ

and . This is the of with respect to .Ð33Ñ \ finite Lebesgue integral 9

86

Remark: The finite Lebesgue integral does not depend on whether the partitions areT!3over the range or the essential range e eÐ\Ñ Ð\ÑÞess

By our assumption of -finiteness of , is separable. For if it were not, there5 A eessÐ\Ñwould exist an uncountable disjoint collection of balls in , withÖF ×! W. 5 .Ð\ ÐF ÑÑ !"

! , contradicting the -finiteness of .For the case of a general -finite measure and measurable function ,5 . A W\ À Ä

we disjointly partition ; we assume , diam , wheree .ess œ T ÐT Ñ T _-T −T 3 3 3

‡3

. .‡ "œ \ . The partition is always assumed at most countable, which is possible sincee 5essÐ\Ñ is separable and -finite. The general Lebesgue integral

P P"

3

( ( ("A e

\Ð Ñ Ð. Ð ÑÑ ´ Ð. Ð ÑÑ ´ \ Ð. Ñ Ð(Þ#$Ñ- 9 . - /9 . / 9 .ess

‡

3

‡

\ ÐT Ñ

is defined using finite integrals on the right, and exists if the sum on the right isindependent of the partition . The above independence condition, which may seemTdifficult to test, is natural; see Theorem 8.4.

We must verify that the general Lebesgue integral coincides with the finite one ifeÐ\Ñ is finite, bounded and separable. To this end, we require

Proposition 7.6: If and the finite Lebesgue integral exists, and ifW W /9 ."‡§ Ð. Ñ

P "'W

W W /9 .# "‡§ Ð. Ñhas positive Borel measure, then exists.'

W#

Proposition 7.7: If the finite integral exists and is a partition of ,P "'W !/9 . WÐ. Ñ T‡

"

then

P

" 5

P( ("W

/9 . / /9 . /Ð. Ð ÑÑ œ Ð. Ð ÑÑß Ð(Þ#%ч ‡

5

‡

T

!

order independently, where !5

‡

5 " #/ / /œ ‡ ‡á Þ

Proof of Proposition 7.6: We will require the fact that the finite Lebesgue integral isalways infinitely divisible (this follows from the definition and a stronger result is provedin Theorem 7.9). Hence the characteristic function of does notF /9 .Ð>Ñ ] œ Ð. Ñ'

W"

‡

vanish. Let be an infinitesimal sequence of partitions of , and a sequence forT T! !Ð#Ñ Ð"Ñ

#W

W W" # 3 3 3 3Ð"Ñ Ð"Ñ Ð#Ñ Ð#Ñ

µ \ − T ß \ − T. Let , and! ! ! !

\ œ \ Ð Ñ Ð4 œ "ß #ÑÞ! ! !4

33 3

Ð4Ñ Ð4Ñ" 9 .

If is unbounded (i.e., has a subsequence converging in law to a measure strictly less\!Ð4Ñ

than 1 as ) for or , then cannot exist. Thus, there is a subsequence of! Ä _ 4 œ " # ]

87

Ö\ ×!Ð"Ñ which converges in law to some probability distribution, and without loss of

generality we reindex so that , where has ch.f. . Thus,\ \ Ê \ \! !!

Ð"Ñ Ð"Ñ

Ä_

Ð"Ñ Ð"Ñ Ð"ÑF

\ ‡\ Ê ] à! !Ð"Ñ Ð#Ñ

hence

\ ‡\ Ê ] ßÐ"Ñ Ð#Ñ!

or in terms of characteristic functions,

F F ÒÐ"Ñ Ð#Ñ

Ä_Ð>Ñ Ð>Ñ ?Ð>ÑÞ Ð(Þ#&Ñ! !

Since , this shows that converges, as does in law. By (7.25),F FÐ>Ñ Á ! Ð>Ñ \! !Ð#Ñ Ð#Ñ

w- is independent of the choice , proving Proposition 7.6. lim!

! !Ä_

Ð#Ñ Ð#Ñ\ T è

Sketch of proof of Proposition 7.7: Note that terms on the right in (7.24) can be wellapproximated by “Lebesgue sums” , where are elements of a!\ Ð Ñ Ö\ ×! !!w ww3 33

‡9 .

subpartition of . Since the approximants can be made to converge order-T!independently to the left side, so can their limits 'T ‡

!5/9 . /Ð. Ð ÑÑÞ è

In this chapter an assortment of distributions will obtain as Lebesgue integrals, as noprior constraints are placed on existence of the mean and variance of the integrand.

The Lebesgue integral is clearly linear, i.e.,

P PP( ( (Ð\ \ Ñ Ð. Ñ œ \ Ð. Ñ \ Ð. Ñß" # " #‡ ‡ ‡9 . 9 . 9 .

if and are independent for each It is also additive, i.e., the integral over a\ \ − Þ" # - Aunion of disjoint sets is the sum (i.e., convolution, in the distribution picture) ofI I" #

the integrals over and .I I" #

Henceforth, all integrals of real-valued functions will be Lebesgue integrals. Clearly,our integral reduces to standard Lebesgue integration when is the identity, and is9 -\Ð Ña point mass for .- A−

Theorem 7.8: Let have 0 mean a.e. , with and \Ð Ñ Ò Ó @ œ Ð\ Ñ. l Ð\ Ñl.- . X . X .' 'A A

# $

finite. Then the Lebesgue integral of exists, and\

P

"#(

A

\Ð ÑÐ. Ð ÑÑ œ RÐ!ß @ÑÞ- . -

Proof: Let and be as above, and let . WeÖT × Ö × \ œ \Ð Ñß œ \ ÐT Ñ! ! ! ! ! !- - . .3 3 3 3 3"

invoke Theorem 3.6, and note that

88

#

5

X .

X .Ò!

!

! !

! !!

$

$

33$

3

33

#3

Ä!´ !ß

Ðl\ l Ñ

Ð Ð\ Ñ Ñ

!!

$#

$#

since the denominator approaches the integral , while the numeratorÐ Ð\ Ð ÑÑ. Ð ÑÑ' X - . -# $#

vanishes, given that . This showssup3 3Ä_

. Ò!!

!

"3

3 3Ä_

\ Ê RÐ!ß @ÑÞè! !!

."#

§7.3 Basic Properties

Definition 7.9: A probability distribution is if to every there/ stable + ß + !ß , ß , ß" # " #

correspond constants and such that+ ! ,

JÐ+ B , чJÐ+ B , Ñ œ JÐ+B ,Ñß" " # #

where is the d.f. of . If , can be chosen to be 0, then is a .J , ß , ,/ /" # scaling stable

Stable distributions, which are intimately connected to the Lebesgue integral, can becharacterized by

Theorem 7.10: : In order that be stable, it is necessary(Khinchin and Lévy, [KL]) /and sufficient that its characteristic function satisfyF

< F # " = (Ð>Ñ ´ Ð>Ñ œ 3 > -l>l " 3 Ð>ß Ñ ß Ð(Þ#'Ñ>

l>lln ( Ÿ

where , , and# ‘ " (− " Ÿ Ÿ "ß ! Ÿ #ß -   !

= (( (

(Ð>ß Ñ œ Þ

Á "

l>l œ "œ tanln

1

1

##

ifif

Corollary 7.10.1: In order that be scaling stable, it is necessary and sufficient that the/logarithm of its characteristic function have the form

<Ð>Ñ œ - l>l 3- l>l ß Ð(Þ#(Ñ>

l>l" #

( (

where and ; if is arbitrary.-   !ß ! Ÿ #ß l- l Ÿ - l l œ "ß - −" # " ##( ( ( ‘tan 1

89

Proof: The scaling stability condition is . Sufficiency is clear,< < <Š ‹ Š ‹ Š ‹> > >+ + +" #

œ

so we prove only necessity. If is stable, is given by equation (7.26). We assume/ <Ð>Ñmomentarily that . Using (7.26), we have( Á "

3 > -l>l " 3 œ" " " " >

+ + + + l>l ## " (

1 Ÿ" # " #

(( ( tan

3 > -l>l " 3 Þ" " >

+ + l>l ## " (

1 Ÿ((

tan

We conclude that or . In either case, the function fits the formÐ3Ñ œ ! Ð33Ñ - œ !# <(7.26). If , a similar argument shows , completing the proof. ( "œ " œ ! è

The proof of the following lemma uses arguments similar to those in Theorem 7.3and its Corollary, and is omitted.

Lemma 7.11.1: Let be independent r.v.'s, and converge orderÖ\ × \3 3 33

!independently. Let be an infinitesimal array of real numbers (i.e., ,Ö+ × + !83 83 Ò8 Ä _

uniformly in ). Then 3 + \ Ê !Þ!3

83 38Ä_

We have the following characterization of Lebesgue integrals.

Theorem 7.11: In order that be the distribution of a Lebesgue integral of an r.v.-/valued function, it is necessary and sufficient that be scaling stable./

Proof: Assume . It is easy to see that then . Let] œ \Ð Ñ Ð. Ð ÑÑ Á ! Ð Ñ !P' - 9 . - 9 $ Ò

$ Ä !W . W" "

‡ be the essential range of and the induced measure on . Since scaling stability\is preserved under sums, there is no loss in assuming is -bounded and finite inW 3" 9

measure. Let be a sequence of positive numbers. Let be a! œ < < < á T! " # <55

partition of into an at most countable collection of sets of measure smaller than ,W"</ 5

with -diameter less than . For , let be a sub-partition of such that given39"5 5 < <

5 5<   < T T5

any , there is at most one element with andT − T T − T T § T< 4 <4 <4 < 45 5 5 5 5 5

< <5 " " 55

.‡ 5 <<4 "ÐT Ñ Á / 4 < 4

" (note changes with and ). For simplicity, we assume such an

element exists for all ; its measure may be 0. Recall that singletons can be< < − T5 </5

sub-divided (along with their measures), for purposes of partitioning; if this were onlyallowed for singletons of positive measure, the following arguments would be somewhatmore technical. Fix as above, and let be constructed in such a way that 4 T<

5< Ÿ<<Š+5 5"

T T \ 4<4 < 4 < 45 5 5" 5 5 "‹+ is non-empty, containing a fixed element, denoted by , for each . For

<   < \ œ \ − T \5 <4 < 4 <4 < 45 5 5 5, choose , where is as above.

" 5 " " 5 "

90

For each with , choose . For , letT − T ÐT Ñ œ / \ − T <   <<4 <4 <4 <45 5 5 < 5 5

< 5.

T œ ÖT − T À ÐT Ñ Á / ×<5‡ 5 ‡ 5 <

<4 <4< . , and consider

W œ \ Ð ÐT ÑÑß Ð(Þ#)Ñ< <4 <45‡ 5 ‡ 5

T −T

"<4 <

5‡

9 .

which is order-independent by definition of the Lebesgue integral. Given , we< ßá ß <! 5

note that since can be made sufficiently large that for .‡ < 5‡<4 5" 5" <ÐT Ñ Ÿ / ß < <   < ß W

is arbitrarily small (i.e., close to the unit mass at 0). This follows by Lemma 7.11.1, andfrom the fact that at most are representative of each of the sets appears in the sumÖT ×< 4

545

(7.28). We thus successively select so that , uniformly in< ß < ßá ÐW ß Ñ !! " !P 5‡

<3 $ Ò5 Ä _

< − Ò< ß_Ñ5" .

We now choose our final partitions. Let

T ´ T ß \ ´ \ ß T ´ T ß W ´ W Ð< − Ò< ß < ÑÑÞ< <4 5" 5#< <4 < < < <5 5 ‡ 5‡ ‡ ‡5

By the above,

W Ê !Þ<‡

<Ä_

Thus

W ´ \ Ð/ Ñ Ê ] œ \Ð Ñ Ð. Ð ÑÑÞ Ð(Þ#*Ñ< <4

T ÂT

<

<Ä_" (<4 <

‡

P9 - 9 . -

A

However, each element of may be divided exactly in half, ,T µ T T œ T T< <4<‡

<4 <4Ð"Ñ Ð#Ñ

and we may choose independent copies of , to be contained in and\ ß \ \ T<4 <4Ð"Ñ Ð#Ñ Ð"Ñ

<4 <

T<Ð#Ñ; respectively. Then

" " Š ‹T ÂT T ÂT

<4 <4Ð"Ñ Ð#Ñ

< </#

< < <<Ä_

<4 <4< <‡ ‡

<

\ \ œ ÐW ‡W Ñ Ê ] Þ/ /

# # Ð/ Ñ9 9

9

9

Thus exists, and in distribution. Similarly, for6Ð#Ñ œ ] ´ ] ‡] œ 6Ð#Ñ]lim$

9 $9 $Ä!

Ð# ÑÐ Ñ

‡#

8 − ß ] œ 6Ð8Ñ] 6Ð8Ñ − ] ‘ <‡8 for some . Thus is infinitely divisible; if is thelogarithm of its ch.f. then

8 Ð>Ñ œ Ð6Ð8Ñ>Ñ Ð8 œ "ß #ßá ÑÞ Ð(Þ$!+Ñ< <

Thus,

" "

7 6Ð7ÑÐ>Ñ œ > ß Ð(Þ$!,Ñ< <

91

and . We define If , let , the8 87 6Ð7Ñ 7 6Ð7Ñ

6Ð8Ñ 6Ð8Ñ3 3< < - - Ð>Ñ œ > 6 œ Þ   ! ; Ä ß ; −Š ‹ ˆ ‰

rationals. Then

-< < <Ð>Ñ œ ; Ð>Ñ œ Ð6Ð; Ñ>ÑÞlim lim3Ä_ 3Ä_

3 3

This holds for all . Thus if is not identically exists, is> − !ß 6Ð; Ñ ´ 6Ð Ñ 6‘ < -lim3Ä_

3

continuous, and . Letting , we have . Thus, if-< < - - ‘Ð>Ñ œ Ð6Ð Ñ>Ñ Ð − Ñ ; Ä ! 6Ð!Ñ œ !3

< !ß œ < ß ! + ß + ! < œ6Ð Ñ6Ð Ñ +" # " #

+--"

# #

" for some . Thus, if , then letting , there exists- -

- ! such that

< < < - < -

- - <

< - -

Ð+ >Ñ Ð+ >Ñ œ Ð-6Ð Ñ>Ñ Ð-6Ð Ñ>Ñ Ð(Þ$"Ñ

œ Ð Ñ Ð->Ñ

œ Ð6Ð Ñ->Ñß

" # " #

" #

" #

so that is scaling stable.]Conversely, assume is a scaling stable r.v. Let (7.27) be its log ch.f. and ] œ Ö] ×A

be a singleton space with measure one. For each , let be independent copies! − Ö] ×!3 3

of in which are assigned measure , and let . Then ] Ð Ñ œ W œ ] Ð ÑA . 9 - - .3 3 33

! ! ! !( ! "

(

has ch.f.

< . .

.

! ! !

( (

!( (

( (

Ð>Ñ œ - > 3- > Ð(Þ$#Ñ>

l>l

œ - l>l 3- l>l ß>

l>l

œ - l>l 3- l>l ß>

l>l

" ¹ ¹ ¹ ¹ "3

" #3 3

3

3 " #

" #

" "( (

so that the distribution of is independent of . Letting also denote the identityW ]! !function on , this shows that in distribution. A 9 .'

A !] Ð. Ñ œ W œ ] èlim!Ä_

We now show that the scaling function must be very restricted for a (non-trivial)9Lebesgue integral to exist. To properly motivate this, we make some observations aboutso-called semi-stable stochastic processes [La]. A stochastic process on is \> ‘ semi-stable if for every with , and denoting+ !ß \ ¶ ,Ð+Ñ\ -Ð+Ñ ,ß - − ¶+> > ‘isomorphism. We assume is continuous, i.e., that\>

lim2Ä!

>2 >c %Ðl\ \ l Ñ œ !ß

and that is proper, i.e. non-degenerate for all . We then have\ >>

Theorem 7.12 [La] À If is semi-stable, proper, and continuous in the above sense,\>

and if has independent increments, then\>

,Ð+Ñ œ + ß!

for some .!   !

92

With the proper interpretation, this may be viewed as a special case of the nexttheorem. The identification depends on the fact that any semi-stable, stationary 0-meanstochastic process with independent increments can be written]>

] œ \Ð>Ñ Ð.>Ñß>_

>( 9

where is a measurable r.v.-valued function and .\ À Ä9 ‘ ‘

Theorem 7.13: If exists and is non-zero, then there exists such] œ \ Ð. Ñ !P'A (9 . "

that for some . Specifically, is given by (7.27), where is the log9 $

$

Ð Ñ"(

Ò ( <$ Ä !

- - !

ch.f. of .]

Proof: Note that convergence of implies . Let . Since] Ð Ñ ! œ Ð\Ñ9 $ Ò W e$ Ä !

" ess

] œ Ð. Ñ Ð œ \ Ñ'W"/9 . . .‡ ‡ "exists , it does also (and is non-zero) on some bounded,

finite sub-domain of Thus assume without loss that is -bounded and -finite.W W 3 ." "‡Þ 9

Let be constructed as in (7.29). Let each contributing to be subdivided into W T W 8< <4 <

equal pieces, each containing (again, arguments become technical if formal\<4

subdivision of is not allowed). This subdivision gives\<4

" " Š ‹T ÂT T ÂT

<4 <4

<‡8 ‡8

<

/8

< <Ä_<4 <4< <

‡ ‡

<

\ œ \ Ð/ Ñ Ê ] ß Ð(Þ$$Ñ/

8 Ð/ Ñ9 9

9

9

so that, letting < Ä _ß

] œ]

6Ð8Ñ

‡8

in distribution, where By (7.27), therefore, . Thus,6Ð8Ñ œ Þ 6Ð8Ñ œ 8lim 9 $9 $Ð8 ÑÐ Ñ

"(

9Š ‹"8 µ -8 - !"

( for some .

We now assume is not asymptotic to , for a contradiction. Assume without9 $ $Ð Ñ -"(

loss that for some sequence , for . Let .$ 9 $ % $ % $8 8 88=Ä !ß Ð Ñ   Ð- Ñ ! œ /

"

8(

Then by the proof of Theorem 7.11,

W œ Ð Ñ \ Ê ] Þ Ð(Þ$%Ñ= 8 = 4

T ÂT8Ä_

8 8

= 48 =8‡

9 $ "We assume without loss that the numbers (see proof of Theorem 7.11) have the<8property that is integral./<8

Let be defined by , where denotes greatest integer. By= / œ Ò/ Ó Ò † Ó8= =8 8

readjustment of in the proof of Theorem 7.11, we assume without loss that andÖ< × =8 8

= Ò< ß < Ñ T Ö\ À \8 5 5" < =4 =4 lie in a single interval . Furthermore, we can choose such that appears in appears in S for . If ˜W × ¨ Ö\ À \ × = Ÿ = Ÿ = \ œ Ö\ À \= = 4 = 4 = 8 8" 8 =4 =48 8 8

93

appears in for some , but not in (we treat all distinct elements ofW = Ÿ = Ÿ = W ×= 8 8" =8

\ as independent), then8

W œ \ Ð/ Ñ Ê !Þ Ð(Þ$&ј8

\−\

=

8Ä_"

8

89

Since the argument for this is similar to one for in the proof of the last theorem, weW8‡

omit it.It follows from (7.34) and (7.35) that

9Ð/ Ñ \ Ê ] Þ=

T ÂT

= 48Ä_

8

= 48 =8‡

8"

This provides the contradiction, since

9Ð/ Ñ \ Ê ] ß Ð(Þ$'Ñ=

T ÂT

= 48Ä_

8

= 48 =8‡

8"

and by assumption fails to converge to 1 as 99Ð/ ÑÐ/ Ñ

=8

=8 8 Ä _Þ è

It follows from Theorem 7.13 and Corollary 7.3.1 that the set of admissible functions9 (i.e., measurable functions yielding non-trivial integrals) fall into equivalence classeswith if for some constant . Furthermore, each class has exactly9 9" #

Ð ÑРѵ - -9 $

9 $"

#Ò$ Ä !

one homogeneous representative . Thus the classes of are indexed by the9 $ $ 9Ð Ñ œ"(

positive reals.

94

Chapter 8

Integrability Criteria and Some Applications

In the first half of this chapter , we introduce an important integrability criterion forrandom variable-valued functions (Theorem 1). We then compare the Lebesgue with theRiemann integral, and finally give applications to the calculation of asymptotic jointdistributions of commuting observables.

A central result shows the non-linear Lebesgue integral of Chapter 7 to becontinuously imbeddable into a standard Lebesgue integral over the space ofM Q‡ ‡

measures on the Borel sets of , the space of probability distributions. Let be the spaceW Vof logarithms of ch.f.'s (log ch.f.'s) of infinitely divisible distributions. Let beN À ÄW Vthe partially defined function , where . Let beN œ Ð/ "Ñ. œ/ / / / .lim

$$ $ $ 9 $Ä!

3>B " BÐ Ñ

' ˆ ‰a measure on the Borel sets of , and denote the integration operatorW VM À Q ć ‡

M œ N . Ð ÑÞ‡ . / . /(Finally, let denote the operation taking a ch.f. to its probability distribution, and Y I ÀV W < YÄ I œ Ð/ ÑÞ be given by <

Theorem 8.5 states that if denotes the Lebesgue integralM À Q Ä9‡ W

M œ Ð. Ð ÑÑß9W

. / 9 . /P(

then the diagram

95

commutes. Specifically, the Lebesgue integral of a r.v.-valued function is\ À ÄA W

P( ( Š ‹A A $

- 9 $\ Ð. Ñ œ Ð/ Ñ " . Ð Ñ Þ"

9 . Y X . -$

exp limÄ!

3\Ð Ñ> Ð Ñ

§8.1 The Criterion

The metric (equation (7.21)) on the space of probability laws will be used3 W9

throughout this chapter. Let be measurable from a space and\ À Ä § Ð ß ß ÑA W W A T ."

. . / / / / / / /‡ "3 3 " # 3 " #

3 3

‡œ \ Þ ß œ ‡ ‡á à œ ÞÞÞFor measures let as usual, let! !We say is - if it is -separable. Note that -Lebesgue integrability isW 9 3 9" separable 9

senseless if or some subset containing the essential part of the range is not -W 9"

separable, since required partitions cannot be countable. A function isT \ À Ä! A Wbounded if its range has finite -diameter. We now show that, in complete analogy to39the real-valued case, if is finite and is bounded and measurable, then it is. A W\ À Ä "

Lebesgue integrable. We require

Lemma 8.1.1: Let , be two countable families of finite Borel measures onÐ+Ñ / /"3 #3

‘. Then

3 / / 3 / / 3 / /P P P

3 3 3

3 "3 3 #3 3 "3 #3 "3 #33

" " "+ ß + Ÿ + Ð ß Ñß Ð ß Ñ Þ Ð)Þ"Ñmax sup

(b) Similarly, if is a measure space and are d.f.'s, thenÐ ß ß Ñ J ÐBÑß K ÐBÑW T .‡/ /

3 . / . /P ‡ ‡ ( (W W

/ /J ÐBÑ. Ð Ñß K ÐBÑ. Ð Ñ

Ÿ ÐJ ßK Ñ. Ð Ñß ÐJ ßK Ñ Þmax sup (W

/ / / //

3 . / 3P ‡ P

Proof of a :Ð Ñ Set , and let be the right side of (8.1). Letting be the% 3 / / %3 "3 #3 83Pœ Ð ß Ñ J

corresponding distribution functions,

" " """

Ð+ J ÐB ÑÑ Ÿ + ÐJ ÐB Ñ Ñ Ÿ + J ÐBÑ

Ÿ + ÐJ ÐB Ñ Ñ

Ÿ Ð+ J ÐB ÑÑ Þè

3 "3 3 "3 3 3 3 #3

3 "3 3 3

3 "3

% % % %

% %

% %

96

For a function from a measure space to a topological space recall that is in\ −A W / Wthe essential range if for every neighborhood of .e . /essÐ\Ñ \ ÐRÑ ! R"

Recall also

) ; ;ÐBÑ œ ß ÐBÑ œ ß œ Þ Ð)Þ#ÑB B

" B " B

Bà lBl Ÿ "

à lBl   "

#

# #‡

"B

œ

Theorem 8.1: Let be finite, and , be bounded and -BorelÐ ß ß Ñ \ À ÄA U . A W 39measurable. Then exists if and only if] œ \Ð Ñ Ð. Ð ÑÑ

P'A - 9 . -

KÐ à BÑ ´ . ÐCÑß ´ . ÐCÑ Ð)Þ$ÑC C

" C " C/ / # / - w lim lim

$ $$ / $

Ä! Ä!_ _

B _#

# #( (exist for every in the essential range of , where . Furthermore, the/ / /\ œ$ $ 9 $

" BÐ ÑŠ ‹

pa r3

# # . - . -œ Ð Ñß KÐBÑ œ KÐ\à BÑ. Ð Ñ( (\

is the Lévy-Khinchin transform of .]

Proof: Assume and exist for supp where . Let # / / / . . .Ð Ñ KÐ Ñ − œ \ ÖT ׇ ‡ " _œ"! !

be an infinitesimal sequence of partitions of For , we first verifyW e /" 5 5œ Ð\ÑÞ − Tess ! !

that the distributions form an infinitesimal sequence. To this end, we" BÐ Ñ Ð Ñ59 . 9 .!

! !5 5/ Š ‹

need to show uniformly in . Suppose this is false. Then" BÐ Ñ Ð Ñ Ä_

!9 $ 9 $ $/ $ / eŠ ‹ Ê − Ð\Ñess

there exists such that for every , there is a , such that% / e ! R ! − Ð\Ñess/ / %( . This contradicts the boundedness (i.e., finite diameter) of_ß RÑ ÐRß_Ñ  

e ) /essÐ\Ñß ÐCÑ. since then can be made arbitrarily large forsup$

$ 9 $" B

_

_

Ð Ñ' Š ‹

/ e− Ð\ÑÞess Thus the sequence is infinitesimal.

According to Corollary 7.3.1, it now suffices to show

" ( 5

5 5_

_5

5, . Ð)Þ%+Ñ

C ,! !

!

!; / Ò #

9 ! Ä _

"( 5 _

B

55

5 Ä_) /

9. Ê KÐBÑß Ð)Þ%,Ñ

C ,!

!

! !

97

where , and9 9 .! !5 5œ Ð Ñ

, œ ÐCÑ. Þ Ð)Þ&ÑC

! !!

5 5_

_‡

5( ; /

9

First consider (8.4b). Note that

" "( ( 5 5_ _

B B,

5 55

5 5) / ) /

9 . 9 .ÐCÑ . œ ÐCÑ.

C , C

Ð Ñ Ð Ñ! !

!

! !

!5

, ÐC , Ñ. Ð)Þ'ÑC

Ð Ñ! !!

!5 5

_

B,w ‡

55

( !5

) /9 .

where . In addition, is bounded uniformly in and (for l, l Ÿ l, l , 5! ! !.5‡

5 5"!5

! !

sufficiently large) for all non-vanishing , since the support of is bounded and. .!5‡

replacing in the metric by gives an equivalent one (see the remark after; 3 ;9‡

Proposition 7.4.). The second term in the sum on the right of (8.6) converges to 0uniformly in and since are infinitesimal, so thatB 5 š Š ‹›/! 9 .5

BÐ Ñ 5!5

3 ) / ) / Ò9 . 9 .

P

5 5_ _

B B

5 55

5 5 " "( (. ß . !Þ Ð)Þ(Ñ

C , C

Ð Ñ Ð Ñ! !

!

! !! Ä _

Let be the d.f. of . ThenK Ð à BÑ œ ÐBÑ. .$ $$ 9 $/ ) / ) /"_

B CÐ Ñ

' Š ‹3 / . / ) /

9 .P ‡

5 _

B

55

( ("W

!!"

KÐ à BÑ. Ð Ñß .C

Ð Ñ

Ÿ KÐ à BÑ. Ð Ñß KÐ à BÑ. Ð Ñ Ð)Þ)Ñ3 / . / / . /P ‡ ‡

5 5T T5 " "( (

! !5 5

!

KÐ à BÑß K Ð à BÑ Þ3 . / . /P

5 5

5 5 5 5 " "! ! ! . !!5

To show (8.8) vanishes as , we note the first term on the right vanishes by! Ä _Lemma 8.1.1, if

3 / /P5ÐKÐ à BÑßKÐ ß BÑÑ !! Ò! Ä _

for , uniformly in and ; the latter follows from the metric infinitesimality of/ /− T 5!5

the partition , and the equivalence of and for fixed .T !! 9 $3 3 $Pß

98

The second term on the right requires some care, however. For , , let% $ !I œ Ö − À ÐK Ð ÑßKÐ ÑÑ Ÿ Ÿ × B%$ $/ W 3 / / % $ $" "

P"

for ; note we have suppressed . LetI œ Ö − À Ð ß I Ñ Ÿ × I − I%$ " 9 %$ %$ %$ "à " à/ W 3 / " " / be the -neighborhood of . Let and$ $ / 3 / / ""

w wŸ − I Ð ß Ñ Ÿ # ß. If , and we have%$ 9

3 / / 3 / / 3 / / 3 / /" % "% "

P P w P w w P wÐK Ð ÑßKÐ ÑÑ Ÿ ÐK Ð ÑßK Ð ÑÑ ÐK Ð ÑßKÐ ÑÑ ÐKÐ ÑßKÐ ÑÑŸ # #œ % à

Ð)Þ*Ñ$ $ $ $" " " "

we have used the definition (7.21) of in terms of Thus3 / / 3 / /9 $ $Ð ß Ñ ÐK Ð ÑßK Ð ÑÑÞ" # " #P

I § I Þ Ð)Þ"!Ñ%$ " % " $à Ð % Ñ

Given , let , diam . By the Lemma (! % $ . "− œ ß œ # œ # ÐT Ñ µ"5 5 5 5! ! !sup sup

denotes complement)

3 . / . /P

5 5

5 5 5 5 " "! ! ! . !KÐ Ñß K Ð Ñ!5

Ÿ ÐT I ÑKÐ Ñß ÐT I ÑK Ð Ñ Ð)Þ""Ñ3 . / . /P ‡ ‡

5 5

5 5 5 5 " "! %$ ! ! %$ . !!5

ÐT I ÑKÐ Ñß ÐT I ÑK Ð Ñ Þ3 . / . /P ‡ ‡

5 5

5 5 5 5 " "! %$ ! ! %$ . !˜ ˜

!5

If , then by (8.10), since diam . HenceT I Á T § I ÐT Ñ Ÿ! %$ ! !% " $"

5 5 5Ð % Ñ #9

/ 3 / / % "! ! . !% " $ 95 5 5Ð % Ñ Ö5ÀT I Á ×P− I ÐKÐ ÑßK Ð ÑÑ Ÿ % !, and , so thatsup

! %$ !5 5Ò! Ä _

the first term on the right of (8.11) vanishes as by Lemma 8.1.1. The second! Ä _ßterm, on the other hand, is bounded by . For ˜. / / / W‡

Ÿ"à − " # "ÐI Ñ K Ð ß_Ñ ß − ß%$ $ / W $sup" " "

K Ð ß_Ñ K Ð ß_Ñ Ÿ Ð Ñ _ß$ $/ / W" # "diam

so that . Since , this term vanishes as so˜sup$ / W

$ %$" "

"Ÿ"à −

‡K Ð ß_Ñ _ ÐI Ñ ! Ä _ß/ . !Ò! Ä _

that (8.8) vanishes as . Together with (8.7) this proves (8.4b) (recall! Ä _' 'AKÐ\ß BÑ. Ð Ñ œ KÐ ß BÑ. Ð ÑÑ. - / . /

eessÐ\Ñ .‡

We now prove (8.4a). By the mean value theorem,

" ( 5

5 5_

_5

5, .

B ,! !

!

!; /

9

œ . , Ð" ÐB , ÑÑ. ßB B" "( (

5 5_ _

_ _

5 5 55 5

w ‡5; / ; /

9 9! ! !

! !!

99

for some By the arguments for (8.6), the last term vanishes as , andl, l l, lÞ Ä _! !‡ !

we are left with proving

"( 5 _

_

55

; / #9

. Ä àB

!!

which also follows along the same lines as the first part of the proof.We now prove the inverse of the above, namely, that (8.4) is necessary for

convergence. Suppose first that for some supp fails to exist. Define / . / %− ß KÐ Ñ !‡

by%

. W3 / /

‡" Ä! ß

P

Ð Ñœ ÐK Ð ÑßK Ð ÑÑÞ Ð)Þ"#Ñlim sup

$ $ $ $$ $

" #

" #

Let . We show that fails to exist, andR œ Ö − À Ð ß Ñ Ÿ × Ð. Ñ/ W 3 / / /9 .w w ‡" % R9

%P'

hence (see Proposition 7.6) the integral over fails to exist. To this end, we mayW"

assume that . Let be an infinitesimal sequence of partitions of . In theR œ TW W" "!

following, we may assume the partitions are even without loss, by theÐ œ Ñ. .! !3 4

arguments of Theorem 7.11 (i.e., the total measure of those elements for whichT!3.‡

3ÐT Ñ! does not equal the common value can be made arbitrarily small). Then (letting$ .! ! be the common value of )5

3 . / . W / 3 $ / $ /P ‡ P

5 5 5

5 5 " 5 " " "! . ! $ ! $ ! ! $K Ð Ñß Ð ÑK Ð Ñ œ K Ð Ñß K Ð Ñ! ! ! !5

Ÿ ÐK Ð ÑßK Ð ÑÑß ÐK Ð ÑßK Ð ÑÑ Ð)Þ"$Ñ

Ÿ Ð Ñß œ ß% % %

max sup "

5

P P5 5

5

‡"

$ 3 / / 3 / /

% % %. W

! $ ! $ $ ! $! ! ! !

max

assuming (without loss) that . W‡"Ð Ñ Ÿ "Þ

Again without loss of generality, we may assume by (8.12) that the sequence is$!such that

. W 3 / / %‡ P"

Ä_ ß  Ð Ñ ÐK Ð ÑßK Ð ÑÑ œ Þ Ð)Þ"%Ñlim sup

! ! ! !$ $

" #" #! !

Hence by (8.13)

lim sup! ! ! !

! . ! ! . !Ä_ ß  

P

5 5

5 5 5 5" #

" 5 " # 5 #" #3 . / . /

% " "K Ð Ñß K Ð Ñ   ß Ð)Þ"&Ñ#! !

so that fails to converge, and by (8.6), so does!5

5 5. /! . !K Ð Ñ!5

100

!5

5 5 5. / !! . ! !K Ð ÐB , ÑÑ Ä _Þ!5

. (The last term in (8.6) still vanishes as ) Thus

P "'W /9 .Ð. ч fails to exist by Corollary 7.3.1.

Now assume fails to exist. If the weak limit of exists, then# ) // $.

( ( (_ _ _

_ _ _‡; / ; / ) /ÐBÑ. ÐBÑ œ ÐBÑ. ÐBÑ 0ÐBÑ ÐBÑ. ÐBÑß Ð)Þ"'Ñ$ $ $

where so that the left side of (8.16) also fails to converge as . From0 − G Ð Ñß Ä !F ‘ $here on the argument using Corollary 7.3.1 is the same as above, and again

P' /9 .Ð. ч

fails to converge. This completes the proof. è

The above conditions can be simplified to a large extent.

Definition 8.2: For denotes those functions defined for small and( ‘ T 9 $ $w − ß Ð Ñ(w

positive satisfying . Let . Define .9 $

$ ( (Ð Ñ

 „

(w w "#

wÒ ‘ T T ‘ ‘$ Ä !

G − œ ´ µ Ö!×-Theorem 8.3: Let and satisfy the hypotheses of Theorem 8.1. Then\ß ß9 .] œ

P'(\ Ð. Ñ9 . exists if and only if

Ð3Ñ −  9 T ((w for some w "#

Ð33Ñ \ for in the essential range of , converges weakly to a measure on / / ‘$„

as $ Ä _

Ð333Ñ − ß Ð+Ñ œ Ð Ñ _à Ð,Ñ Ÿ "ß Ð Ñ œ !for supp if then if ; and/ . ( i / ( X /‡ w w" "# #

Ð-Ñ œ "ß B . if exists.( /wÄ_ Q

Qlim( 'Note that the weak convergence condition above means converges to (possibly/$

infinite) measures on and , individually.‘ ‘

Proof: We first show the above imply the necessary and sufficient conditions of theprevious theorem. Let w- w- . Simple scaling argumentsU ´ ´/ / /lim lim$ $ $ $Ä!

" BŠ ‹(w

(replace by and let ) show that is homogeneous, and thatB -B Ä ! U$ /

.ÐU Ñ œ - lBl - lBl .B ÐB − Ñß Ð)Þ"(ÑB

lBl/ ‘ " #

" " „( (

where . For , let . Then since( / /œ B ! J ÐBÑ œ ÒBß_Ñà J Ð BÑ œ Ð _ß BÑ" (w

101

" B - -J B ß$ $ (

Ò " # ((

w $ Ä !

it follows that

B J ÐBÑ ß- -( " #

Ò(B Ä _

and similarly

ÐBÑ J Ð BÑ Þ Ð)Þ")Ñ- -( " #

Ò(B Ä _

If , then for supp ( / .w ‡"# − ß

( (" Q

" Q# # #B . œ Q B . ÐBÑß Ð)Þ"*Ñ/ /$

(

where (8.19) is easily shown to converge using the asymptotics of and ,Q œ à J J" $

after integration by parts. Hence converges weakly on and thus B . Ò "ß "Ó ÐBÑ.# / ) /$ $

does on , so that the first weak limit in (8.3) exists. To examine the same limit when‘( i /w "

#œ Ð Ñ, notice that (8.19) now converges by the finiteness of .We prove existence of the second limit in (8.3). If , the limit follows from the(w   "

convergence of

( (" Q

" Q"B . œ Q B. ÐBÑß Ð)Þ#!Ñ/ /$(

via integration by parts as before. Then ,"#

wŸ "(

Q B. œ Q B. B. ß( (" "

Q Q _

Q _ Q( ( (– —/ / /

and the latter again converges via integration by parts. This completes the proof ofsufficiency.

We now assume (8.3) to hold, and prove - ( is clearly necessary byÐ33Ñ Ð333Ñ Ð3ÑTheorem 7.11). First, follows immediately from convergence of , since weakÐ33Ñ KÐ à BÑ/convergence of on implies the same for . To prove , first let .) / ‘ / (. . Ð333Ñ œ$ $

„ w "#

Since converges weakly by hypothesis,) /. $

lim lim$

$Ä! " Q

" Q# #

QÄ_( (B . œ B ./ /

converges, proving (a). Similarly, if , thenÐ333Ñ Ÿ ""#

w(

102

lim lim$

$(

Ä! " Q

" Q

QÄ_

"( (B. œ Q B. Þ Ð)Þ#"Ñ/ /

Since this converges (by 8.3) and it follows that . To( / " !ß B . œ !limQÄ_ Q

Q'showthat indeed has a first moment, we show (since the argument on is/ / ‘'

!_ B . _

the same). Let be a monotone function, , defined by . By (8.19)K KÐ!Ñ œ ! .K œ B./and (8.3), we have convergence as ofQ Ä _

Q B . œ Q ÐKÐQÑ EÐQÑÑß( (# # "

!

Q( /

where

EÐQÑ œ KÐBÑ.BÞ Ð)Þ##Ñ"

Q(!

Q

Hence

KÐQÑ EÐQÑ Ÿ 2ÐQÑ ÐQ !Ñß

where is a smooth positive function, with 2ÐQÑ 2Ð!Ñ œ !ß 2 Ð!Ñ Ÿ G ßw"

2ÐQÑ œ G Q Q   " KÐBÑ Ÿ B#"( for ; note that . There exists a monotone function

K ß K Ð!Ñ œ !‡ ‡ , satisfying

K ÐQÑ E ÐQÑ œ 2ÐQÑ߇ ‡

where and are related by (8.22). The equation for ,K E Q   "‡ ‡

K ÐQÑ E ÐQÑ œ G Q Ð)Þ#$ч ‡ "#

(

is solved by differentiating once, giving

K ÐQÑ œ G G Q ÐQ   "Ñß#

" ‡

$ # (( (

so that is bounded. On the other hand,K‡

ÐK KÑ ÐE EÑ   ! Ð)Þ#%ч ‡

on this implies for , so is bounded, proving‘ ‡ ‡à ÐK KÑÐQÑ   ÐK KÑÐ!Ñ Q   ! Kthat has a finite first moment on . The proof is of course the same on . Thus/ ‘ ‘

necessity of (b) is proved; necessity of (c) for convergence of the secondÐ333Ñ Ð333Ñintegral in (8.3) is clear, and this completes the proof. è

We now consider the general (non-finite) Lebesgue integral on a measure spaceÐ ß ß ÑA U . . Recall (8.2).

103

Lemma 8.4.1: Let be a sequence of independent r.v.'s, and . The sum\ , œ Ð\ Ñ3 3 3‡X;

! ! !3œ"

_

3 3 3 3\ l, l Ð\ , Ñ converges order-independently if and only if and converge.X)

Proof: Assume the numerical series converge absolutely, and write, using the mean valuetheorem,

" " "3 3 3

3 3 3 3 3w ‡

3X) X ) X )Ð\ , Ñ œ Ð\ Ñ , Ð\ , Ñß Ð)Þ#&Ñ

where with probability one. It is easy to see the are infinitesimal, and inl, l Ÿ l, l \3‡

3 3

particular that , so by (8.25) converges. From this itX ) Ò X )w3 3 3

3Ä_Ð\ , Ñ ! Ð\ Ñ!

follows (with convergence of ) that converges as well, and sufficiency! !l, l Ð\ Ñ3 3X;

follows by Lemma 7.3.1. Conversely, using Lemma 7.3.1, if and ! !X ) X;Ð\ Ñ Ð\ Ñ3 3‡

converge absolutely, so does , and converges by (8.25). ! !X; X )Ð\ Ñ Ð\ , Ñ è3 3 3

Recall that, for a measure on and , the measures and are/ ‘ 9 ‘ ‘ / /À Ä K $

/ / / ) /$ 9 $

$ $$

œ ß .ÐK Ñ œ . ß Ð)Þ#'Ñ" B

Ð Ñ w- limÄ!

and

# ; / # ; // $ $$ $

/œ ÐBÑ. ÐBÑß œ ÐBÑ. ÐBÑ Ð)Þ#(Ñlim limÄ! Ä!_ _

_ _‡ ‡( (

(see (8.2)), when the limits exist. If is an r.v., and are defined analogously by\ K\ #\the distribution of . If is a measure, denotes total measure./ . .\ l l

Theorem 8.4: Given : , and , a random variable-valued function9 ‘ ‘ A W Ä \ À Äon the measure space , the integral exists if and only if A - 9 . -] ´ \Ð Ñ Ð. Ð ÑÑ Ð"ÑP '

A

K \ Ð#Ñ lK\Ð Ñl. Ð Ñ/ # / - . - and exist for every in the essential range of , and and/ A '' ' '¹ ¹A A A-# . - # . .\Ð Ñ \. Ð Ñ Ð . ß K\. Ñconverge. Furthermore, the pair is the Lévy-Khinchin transform of .]

Note that we make no restriction on , i.e., may be measurable map into theW \ anyspace of probability distributions. This theorem, together with Proposition 7.6, showsthat an integrable induces a countably additive measure-valued measure defined by\ `` 9 .ÐFÑ œ \ Ð. Ñ'

F .

Proof: Assume (1) and (2) hold. Let be the measure induced on by . Since. W‡ \

l l Ÿ - l l - lK l - ß - !ß .# # / # ./ / W -‡ ‡ ‡

" # " # \Ð Ñ for some converges as well. Let' ¹ ¹

104

ÖT × œ Ð\Ñ3 3−M " be an at most countable partition of into sets of finite measure andW eessdiameter. Let

W / W # W / W #"" " "# "‡ ‡œ Ö − À   !×ß œ Ö − à !×Þ/ /

Assume without loss that respects the partition of . Let be an atT ß ÖT ×W W W"" "# " 34 4

most countable partition of whose elements have finite measure and diameter, withT3

/ . . 9 9 . ; /34 34 34 34 34 34 34 34‡ ‡

__ B− T ß œ ÐT Ñß œ Ð Ñ , œ ÐBÑ.. Let . By the proof of' ˆ ‰

934

Theorem 8.1, can be made to differ from by arbitrarily little; hence! '4

34 T‡ ‡l, l l l. Ð Ñ

3# . //

the sub-partitions can be chosen so that converges. Given , for aT l, l 3 − M34 343−Mß4

!sufficiently small sub-partition of (still denoted by is unchanged),ÖT × ÖT ×à T34 4 34 3! '4

‡

34 34 T‡/ 9 /9 . / can be made arbitrarily close to . Hence by Theorem 1,

P 3Ð. Ð ÑÑ )Þ

" ( ( 4 _ T

_

3434

34

‡) / / .9

. lK l. Ð)Þ#)ÑB ,

3

can be made arbitrarily small. Therefore by condition (2), there exists a sub-partitionÖT × , . T34 3ß4 34 34 34

3ß4 3ß4__ B, such that and converge absolutely under , as well! !' Š ‹) / 34

349

as any finer partition. Thus, by the Lemma, converges order! !Š ‹3 4

‡ ‡

34 34/ 9

independently. Since the inner sum approximates arbitrarily well, the sumP 3'T

‡/9 .Ð. Ñ! ' '3

‡

T‡

P P3/9 . - 9 . -Ð. Ñ œ \Ð Ñ Ð. Ð ÑÑA is also order independent. Since any two

partitions have a minimal common refinement, this sum is also independent of ÖT ×Þ3

Conversely, if exists, we claim . For if the latter isP' 'A A\ Ð. Ñ lK\l. _9 . .

false, let be an at most countable partition of into sets of finite measure andÖT ×3 3−M A

diameter. The sum diverges; and therefore must also fail! !' '3 3

T3 3lK\l. \ Ð. Ñ. 9 .3

to converge by the linearity and continuity of the Lévy-Khinchin transform (Remarksafter Theorem 7.1). Thus the integral fails to exist. A similar contraction obtains if'A /l l. œ _# .‡ .

Finally, if exists, then , where] œ \ Ð. Ñ ] œ ]P 33

' !A 9 .

] œ \ Ð. Ñ3T

P

3

( 9 .

and is a sequence of subsets of with finite measure and diameter. Thus, the Lévy-T3 AKhinchin transform of is the sum of those of . This together with the final assertion] ]3

in Theorem 8.1 completes the proof. è

105

We remark that conditions (1) can naturally be replaced by conditions - ofÐ3Ñ Ð333ÑTheorem 8.3.

The following lemma is proved in Loève [Lo, §23].

Lemma 8.5.1: Let , and be finite Borel measures on . If! ‘ G ‘8 8−

3> / " .! G8 8_

_ 3>B 3>B"B B

Ð"B Ñ' ˆ ‰# #

#

converges to a function continuous at the origin,then converges and converges weakly.Ö × Ö ×! G8 8 8 8

Theorem 8.5: The integral exists if and only if] œ \ Ð. ÑP'A 9 .

Ð3Ñ Ð>Ñ œ Ð Ð Ð Ñ>Ñ "Ñ > œ ! − Ð\Ñß< F 9 $ / e/ /$ $limÄ!

" exists and is continuous at for ess

where is the characteristic function of F //

Ð33Ñ the integral

< < . -Ð>Ñ œ Ð>Ñ. Ð Ñ Ð)Þ#*Ñ(A

-\Ð Ñ

converges absolutely.

In this case has characteristic function ] /<.

Proof: If exists, then according to Theorem 8.4, so do and for ] K − Ð\ÑÞ# / / e/ essFurthermore, if / e− Ð\Ñßess

lim$

$ /Ä!

3>B 3B># #

#( ( Ð/ "Ñ. ÐBÑ œ 3 > / " .ÐK ÑÐBÑ3B> " B

" B B/ # /

is clearly continuous at 0, proving . According to (7.1), has the log ch.f.Ð3Ñ ]

< #"_

_3B>

# #

#

Ð>Ñ œ 3 > / " .K Ð)Þ$!Ñ3B> " B

" B B(

where, by Theorem 8.4,

# # . / .œ . ß K œ K . ß( (W W

/‡ ‡

and is the induced measure on . Since the integrand in (8.30) is bounded, we can. W‡

interchange integrals, so

106

< # / . /

/ / . /

$

"_

_3B> ‡

# #

#

Ä! _ _

_ _

# #3B> ‡

Ä!

Ð>Ñ œ 3 > / " .ÐK ÑÐBÑ . Ð Ñ3B> " B

" B B

œ 3> . / " . . Ð ÑB 3B>

" B " B

œ"

( (– —( ( (– —( –

W/

W $$ $

W $

lim

lim ( —_

_3B > ‡Ð/ "Ñ. ÐBÑ . Ð Ñ œ Ð>ÑÞ

Ð)Þ$"Ñ

9 / . / <

It is clear that the integrals converge absolutely.Conversely, assume that and hold. If Ð3Ñ Ð33Ñ − Ð\Ñß/ eess

lim lim$ $

$ $ $Ä! Ä!

3B> 3>B# #

#( ( ( Ð/ "Ñ. œ 3> . / " ÐBÑ.3>B " B

" B B/ ; / ) /

is continuous at 0, so that by the Lemma, and w- exist.# ; / / ) // $ $œ . .K œ .lim lim'Let be a countable partition of into sets of finite measure, on each of which ÖT × \3 3−M Ais essentially bounded. This can be done by appropriately partitioning . LeteessÐ\ÑA 9 .8 33œ"

8œ T \ Ð. Ñ 8- '. By Theorem 8.4, exists for all , and by part one of thisA8

proof, if is its log ch.f., then<8

< < . -8 \Ð Ñœ . Ð ÑÞ Ð)Þ$#Ñ(A

-8

Clearly, converges weakly as to an r.v. with ch.f. . Thus is'A8\ Ð. Ñ 8 Ä _ ] Ð>Ñ ]9 . <

independent of the choice of , and is the desired integral. ÖT × è3 3−M

§8.2 The Riemann Integral

Let be a metric space with -finite Borel measure , and be anÐ ß Ñ \ À ÄA 5 5 . A Wr.v.-valued function. The Riemann integral of is defined with respect to partitions of \ Arather than . We might again try to duplicate the elegance of the Lebesgue theory in theWRiemann integral, but we approach the latter with a view to utility, namely, to physicallymotivated applications. With this intent, we define the Riemann integral using globalpartitions of , rather than patching integrals over sets of finite measure whose rangeAunder is bounded. The former definition will dovetail with that of the -integral.\ V‡

Definition 8:6: Let be an infinitesimal sequence of partitions of . Select aÖT ×! !œ"_ A

function : . Let , and assume9 ‘ ‘ - 3 3Ä − T! !

"3

3 3Ä_

\Ð Ñ Ð Ñ Ê ] ß- 9 .! !!

where , and the are independent. If is independent of { },. . -! ! !- A3 3 −œ ÐT Ñ Ö\Ð Ñ× ] T

107

then is the - of (with respect to the non-linear measure ),] Ð Ñ \ Ð. Ñ9 9 .Riemann integral

] œ \Ð Ñ Ð. Ð ÑÑ Þ Ð)Þ$$ÑV(

A

- 9 . -

Theorem 8.7: If is -Riemann integrable then it is -Lebesgue integrable, and the\Ð Ñ- 9 9integrals coincide.

Proof: Recall (7.21). We begin by assuming is finite and is -bounded. Let be. 3\ T9 !

an infinitesimal net of partitions of . Recall the Lebesgue integral may be equivalentlyAdefined by allowing partitions of to subdivide individual elements, even ones ofWmeasure 0. Under this more general (but equivalent) definition, there exists a partitionsequence of such that . As before, we may use the arguments ofT \ÐT Ñ œ T! !!

‡ ‡3 3W

Theorem 7.11 to assume (without subsequent loss) that the partitions are evenT ÐT Ñ! !‡

with respect to , i.e., ; again the measure of those. . . . .Ð œ \ Ñ ÐT Ñ œ ÐT ч "! !3 4

elements with odd (unequal) measure can be made arbitrarily small by taking sufficientlysmall sub-partitions. By taking sub-partitions, we may also assume that vanish inT!3

‡

diameter uniformly as .! Ä _

We proceed by contraposition, assuming the Lebesgue integral fails to exist. In thiscase it suffices to show the Riemann integral fails to exist for any '

A"\ Ð. Ñ §9 . A A"

with positive measure, by arguments used in proving Proposition 7.6. By our assumptionand Theorem 8.1, either or fails to exist for some . Assume does notK − K/ # / e // essexist. At this point, using the above net of partitions, the argument becomes exactlyT!

‡

analogous to that after (8.12), so we omit the details. The same argument works if #/fails to exist, yielding the result when is finite and is -bounded. The fact that the. 3\ 9

two integrals coincide in this case follows from correspondence of the Riemann sumsover the partitions and .T T! !

‡

If is infinite or is unbounded, and exists, then the integral also. A 9 .Ð Ñ \ \ Ð. ÑV'A

exists over any subset with positive measure. Since the essential range of is -A A 53 § \finite, it is separable. Let be an at most countable partition of , with andÖ × Ð ÑW W . W3 3

‡

diam . If , then exists, and we must show that Ð Ñ _ œ \ Ð Ñ \ Ð. ÑW A W 9 .3 3 3"

3P 3' !A

P V3

' 'A \ Ð. Ñ \Ð. Ñ9 . . converges order-independently to . This follows from the fact

A

that can be approximated arbitrarily well by Riemann sums over partitionsP 3'A \ Ð. Ñ9 .

of whose total sum over approximates A 9 .3 3 \ Ð. ÑÞ èV'A

Recall that for ,/ W−

K Ð à BÑ œ ÐCÑ. à Ð Ñ œ ÐCÑ. Þ" C " C

Ð Ñ Ð Ñ$ $/ ) / # / ; /

$ 9 $ $ 9 $( ( _ _

B _

We now prove

108

Theorem 8.8: Suppose

Ð3Ñ \ À Ä is Lebesgue integrable and -continuous,A W 39

Ð33Ñ Ð Ñ ´ K Ð\Ð Ñß_Ñ l Ð\Ð ÑÑl P Ð Ð ÑÑ the function g is in $ % $ $$ $5 - - %

ß " "ŸÐ ß ÑŸ

"- - # - . -sup"

"

" "

for some , .% $ !

Then is Riemann-integrable.\

Proof: We prove this for partitions which do not subdivide points (the proof thereforedoes not directly work for measure spaces with atoms). The general case (which allowspartition elements which overlap on atoms) follows with small modifications.

Let be an infinitesimal net of partitions of , and . LetT − T! ! !A - 5 5

\ œ \Ð Ñß œ ÐT Ñß œ Ð Ñ! ! ! ! ! !5 5 5 5 5 5- . . 9 9 . . We first verify that the double sequenceÖ \ ×9! !5 5 is infinitesimal. Suppose it is not. Taking a subsequence if necessary, we mayassume without loss of generality that

. %! . !5 5 "! ! !!K Ð\ ß_Ñ  

5

for some , and some choice of for each . Therefore if ,% ! $ %" ! 5 !ß!

1 Ð Ñ   Ð Ð ß Ñ Ÿ Ñ Ð)Þ$%Ñ$ % !!

ß 5"

5- 5 - - %

%

.!

!

for sufficiently large that . Furthermore, again taking subsequences if! . $!5! Ÿnecessary, we assume that . Then if (the -ball about. . - %! ! % !!5 5

"# Ð "Ñ5! !!

Ÿ F œ F Ð Ñ"

-!5!),

( ( ( ( A

$ % $ %1 . Ð Ñ   á 1 .ß ßF F µF F µÐF F Ñ

. - ." # " $ " #

  ÐF Ñ ÐF Ñ ÐF Ñ á

  ÐF Ñ ÐF Ñ ÐF Ñ á" "

# #

œ _ß

% % % % %

. . . . .. . .

% % %

. . .. . .

" " " " "

"5 #5 "5 $5 #5" # $

" " "

"5 #5 $5" # $

" # " $ #

" # $

since for sufficiently large; this gives the desired contradiction. Thus,. . !ÐF Ñ  ! !5!

Ö \ ×9! !5 5 is infinitesimal. To prove the theorem, according to Corollary 7.3.1, it suffices to prove the

convergence

109

"( 5 _

B

55

5 Ä_) /

9. Ê KÐBÑà Ð)Þ$&+Ñ

C ,!

!

! !

"5

5Ä_

w, ß Ð)Þ$&,Ñ!!Ò #

for some a multiple of a d.f., and some , both independent of .K À Ä − ÖT ב ‘ # ‘w!

Here, is the distribution of , and/! !5 5\

, œ . ß Ð)Þ$'ÑB

! !!

5 5_

_‡

5( ; /

9

with given by (7.9). We have;‡

"( ( 5 _

B

55

5) / - . -

9. œ L Ð à BÑ. Ð Ñß

C ,! !

!

! A

where

L Ð à BÑ œ . Ð − T ÑÞ" C ,

! ! !! !

!- ) / -

. 95 5_

B

5 55(

Since is -continuous, if supp , then so that by (1) of\ − Ð Ñ \Ð Ñ − Ð\Ñß3 - . - e9 essTheorem 8.4,

L Ð à BÑ Ê LÐ à BÑß Ð)Þ$(Ñ!!

- -Ä_

where is a multiple of a d.f. in for each . Furthermore,LÐ à BÑ B- -L Ð à BÑ Ÿ L Ð à_Ñ! !- -

œ . ÐBÑ. Ð − T Ñß Ð)Þ$)Ñ" C , C

. 9 . 9) / ) / -

! ! ! !! ! !

!

5 5 5 5_ _

_ _

5 5 55‡

w( ( where is determined by the mean value theorem. The first term on the rightl, l Ÿ l, l! !5

‡5

is clearly dominated by for sufficiently large. Since the double1 Ð Ñ − P Ð Ñ$ %ß"- . !

sequence is infinitesimal and , the second term on the right is/ )! 95B wˆ ‰!5

Ð!Ñ œ !

eventually dominated by

L Ð à_Ñ œ œ . Ÿ O 1 Ð Ñ Ð − T Ñß Ð)Þ$*Ñ, " B

!!

! ! !! % $ !

Ð#Ñ ‡5

5 5 5_

_

5 ß 5- ; / - -. . 9

( for sufficiently large, with independent of Thus, for sufficiently large,! ! !O Þ

110

L Ð à ?Ñ Ÿ Ð" OÑ1 Ð Ñ − P Þ Ð)Þ%!Ñ! $ %- -ß"

By (8.37), if 2ÐBÑ − G Ð Ñß-_ ‘

( (_ _

_ _

L Ð ß BÑ2ÐBÑ.B LÐ ß BÑ2ÐBÑ.Bß! - Ò -! Ä _

so that by (8.40), the dominated convergence theorem, and the Fubini theorem,

( ( ( (_ _

_ _

Ä_A A

!!

L Ð à BÑ. Ð Ñ ÐBÑ.B LÐ à BÑ. Ð Ñ ÐBÑ.BÞ Ð)Þ%"Ñ- . - 9 Ò - . - 9

Since is arbitrary, it follows that2 − G Ð Ñ-_ ‘

"( ( ( 5 _

B

55

5) / - . - Ò - . -

9. œ L Ð à BÑ. Ð Ñ LÐ à BÑ. Ð Ñà

C ,! !

!

! A A! Ä _

note that the limit is independent of the choice .T!

Similarly, by (8.39),

" ( (5

5Ð#Ñ Ð#Ñ, œ L Ð à_Ñ . Ð Ñ L Ð à_Ñ. Ð Ñß!

A A! - . - Ò - . -! Ä _

where , completing the proof of (8.35). L œ L èÐ#Ñ Ð#Ñlim!Ä_

!

Note that the proof above holds if is replaced by# /$Ð Ñ

# / ; /$ $"

_

_

"Ð Ñ œ ÐBÑ. ÐBÑß(where is bounded. Thus we have; ;"

#ÐBÑ œ ÐBÑ SÐB Ñ ÐB Ä !Ñ

Corollary 8.8.1: The statement of the Theorem holds if in is replaced by Ð33Ñ Þ# #$ $"

§8.3 The -IntegralV‡

Let be a metric space with -finite Borel measure . Let , andÐ ß Ñ À ÄA 5 5 . 9 ‘ ‘

39 be the corresponding metric on . Let be -measurable with a rangeW A W 3\ À Ä ß9

consisting of independent r.v.'s. We now consider the general version of §3.1, i.e., theresult of summing “samples” of at an asymptotically dense set of points in . In\ AChapter 9, will be the spectrum of a von Neumann algebra of physical observables.A

For , let be at most countable. Note the elements of % A - A A ! œ Ö × §% %4 4−N%

need not be distinct. For , let , where denotes cardinality.K § R ÐKÑ œ l Kl l † lA A% %

We assume that for any open set Ð3Ñ K § ßA

111

% Ò .R ÐKÑ ÐKÑß Ð)Þ%#+Ñ%%Ä!

and for some , if is an open ball with Ð33Ñ G Ð Ñ F ÐFÑ   Gß. A .

% .R ÐFÑ Ÿ 5 ÐFÑ Ð)Þ%#,Ñ%

for some fixed These conditions should be compared with (3.1).5 − ‘.

Definition 8.9: The net is a - of points. We defineÖ ×A .% %! net

V‡( "A %

%\Ð Ñ Ð. Ñ ´ Ð Ñ \Ð Ñß Ð)Þ%$Ñ- 9 . 9 % -limÄ!

4

4

if the right hand limit (in law) is independent of the choice of .Ö ×A% %!

We proceed to relate the to the Riemann integral.V‡

Definition 8.10: Let be a metric space, and . Let diam , and Ð ß Ñ I § 6 œ I =   !A 5 Abe the supremum of the diameters of all balls . The pair are the F § I Ð=ß 6Ñ dimensionsof .I

Lemma 8.12.1: Given , there is a partition of such that each has% A ! T T − T! ! !5

dimensions where . Furthermore, can be chosen so thatÐ=ß 6Ñ =   ß 6 Ÿ & T% % !

.Ð`T Ñ œ ! 5!5 for all .

Proof: Let be a maximal set of disjoint -balls in , and be the collection of centersU % A Gof . For , let be the set of points closer to than to any other˜F − F Ð Ñ − FÐ Ñ −U - U - A -%

point in . Let . If , then the dimensions of satisfy˜ ˜˜ ˜ ˜G œ ÖFÐ Ñ À − G× F − Ð=ß 6Ñ FU - - UÐ=ß 6Ñ § Ð# ß % Ñ =   # ß 6 Ÿ %% % % %, i.e. . There are at most a countable number of elementsof with non-null boundaries. Let be an enumeration of this collection. For˜ ˜ ˜U F ß F ßá" #

( A - A 5 - ( ( !ß K § F ÐKÑ œ Ö − À Ð ßKÑ × Ÿ, let . There exists such that(%

" %

F ÐF Ñ œ Ö À Ð ß Ñ − F ×("˜ ˜ for some has null boundary, since otherwise " " " " "- 5 - - ( - .

would not be -finite. We replace by and decrease the remaining sets in ˜ ˜ ˜5 UF F ÐF Ñ" "(

accordingly. We continue in this manner by then replacing by , and,˜ ˜F F ÐF Ñß Ÿ# # # )(%

#(

in general, replacing by , at each stage adjusting as well. At the˜ ˜ ˜F F ÐF Ñß Ÿ5 5 5 #(%

5 5"( U

end of this process, all sets have null boundary, and dimensions .˜ ˜F − Ð=ß 6Ñ § Ð ß & ÑU % %

We let . ˜T œ è! U

Definition 8.11: A Borel measure on a metric space is if for , every ball ofuniform % !radius has measure bounded below by some constant . A sequence of r.v.'s% - ! Ö\ ×% 3

is if the corresponding d.f.'s satisfy , where and are d.f.'s. Itbounded J J Ÿ J Ÿ K J K3 3

is if it is not bounded. A collection of sets is a of a set if itunbounded partial partition Asatisfies all the requirements of a partition, except possibly . In particular,-

5 5T œ A

112

elements of may be apportioned to several partition elements, if non-zero measures areAdivided correspondingly.

Note that a sequence of r.v.'s is unbounded if for all there exists \ Q   ! !8 %such that .sup

8 "8T Ðl\ l QÑ   %

Theorem 8:12. Let be a metric space with uniform -finite Borel measure . IfÐ ß ÑA 5 5 .\ À Ä VA W is Riemann-integrable, then it is -integrable, and the two integrals‡

coincide.

Proof: Let be an infinitesimal sequence of partitions of , such that (see LemmaÖT ×! !w A

8.12.1) has dimensions dim , and . Since is uniform,T T § ß Ð`T Ñ œ !! ! !! !5 5 5w w w" &Š ‹ . .

there is a function such that . There exists a subpartition of2Ð Ñ ! ÐT Ñ 2Ð Ñ T! . !! !5w

T!w , consisting of sets with null boundary, with

2Ð Ñ Ÿ ÐT Ñ Ÿ #2Ð ÑÞ Ð)Þ%%Ñ! . !!5

If is non-atomic, can be constructed as a standard subpartition of . If hasA T T T! ! !5w

atoms we can write, according to previous conventions, as a union of copies ofT 8!5w

T ÐT Ñ T! ! !5 5w w"

8, each with measure , with each a distinct element of (this is a quick way.

of eliminating the problem of atoms, although the end result could be accomplished bysubdividing only atoms).

Consider a -net satisfying. A -% %w w

4 4œ Ö ×

R ÐT Ñ œ ß Ð)Þ%&Ñ% !!w

55– —".

%

where , and [ ] is the greatest integer function. Given " A ! !ß R ÐT Ñ œ l T l †% %! !w w

5 5

and , subdivide into the partition , where each element satisfies% T ÖT ×! !5 53 3

. . . - A! ! ! ! !.

%53 53 53 53 53R ÐT Ñwœ œ ÐT Ñ T −!

% !

5w

5, where , and contains exactly one element .

If , then" -œ "ß \ œ \Ð Ñ! !53 53

! ! Œ” ! Œ”

5ß3 5ß353 53 53 5

"

5ß44 5

"

\ Ð Ñ œ \

œ \Ð Ñ

! ! ! !

% !

9 . 9 .

- 9 .

.%

.%

!

!

5

5

••

Ð)Þ%'Ñ

where in the last sum is a function of defined by . If increases to5 4 − T œ Ð Ñ- ! ! %% !4 5

infinity sufficiently slowly as , then by (8.45) and the Riemann integrability of ,% Ä ! \if \ œ \Ð Ñß% %4 4-

113

" – — (5ß4

4 55

"

Ä!\ Ê \ Ð. ÑÞ Ð)Þ%(Ñ% !

!

% A

9 . 9 ..

% V

Let denote the greatest integer, be fixed, be an indexingÒ † Ó Ö\ ×! %53 3

Ö\Ð Ñ À − T ×- -% % !4 4 5 , and

+ œ Ð ÑÞ% !!

5 55

"

9 . 9 %.

%– — To prove that

"5ß3

5 53Ä!

+ \ Ê !ß Ð)Þ%)Ñ% %%

we assume that it is not true. Recall that a space of finite measures with bounded totalmeasure is compact in the topology of weak convergence. Thus there exists a sequence% %j jÒ

j Ä _! j Ð3Ñ such that (replacing by )

"5ß3

j5 j53jÄ_

+ \ Ê Ð)Þ%*+Ñ/

for some , or / $Á Ð33Ñ9

"5ß3

j5 j53+ \ j Ä _ Ð)Þ%*,Ñ is unbounded as .

We assume , since the argument is similar otherwise. By Theorems 7.13, 8.7, and ourÐ3Ñ

hypotheses, is homogeneous of positive order, so that is well-defined, and9 9%

"j5

j

Ð+ Ñ

vanishes uniformly as .j Ä _Therefore

– —.

%9 Ò

!5

j

"j5Ð+ Ñ !ß

j Ä _

uniformly in . Thus we may assume (by taking an -subsequence if necessary) that5 j

"– —j

j

j

"j5

.

%9 !

!Ð+ Ñ Ÿ 2Ð Ñß Ð)Þ&!Ñ

for each . For let be a partial partition of , into 5 j œ "ß #ßá ß ÖT × ´ T! ! !.%53 5

j j3 5c ’ “!5

j

subsets, each of measure , with . By (8.50), can be chosen so9 c" j jj5 j53 53 5Ð+ Ñ \ − T! !

that is a partial partition of . By choosing an -subsequence, we mayc c! !!5 5j 5jœ T j-

assume by thatÐ3Ñ

114

3 /P

3ß5

j5 j53 j " + \ ß Ÿ ß"

G

where may be arbitrarily large. Therefore, the partial Riemann sum G ! + \!3ß5ßj

j5 j53

becomes unbounded, contradicting Riemann integrability of ; this proves (8.48). Thus\by (8.47),

9 % 9 .Ð Ñ \ Ê \ Ð. ÑÞ Ð)Þ&"Ñ" (4

4Ä!

%% A

V

By an exactly parallel argument, if in (8.45) , and ," " ! ! ! %œ Ð Ñ ! œ Ð Ñ !Ò Ò! %Ä _ Ä !

the latter sufficiently slowly, then

9 %Ð Ñ \ Ê !Þ Ð)Þ&#Ñ"4

4Ä!

%%

Combining (8.51) and (8.52), if , and" !Ð Ñ !Ò! Ä _

– — – —Ð" Ñ Ð" ÑŸ R ÐT Ñ Ÿ ß Ð)Þ&$Ñ

" . " .

% %! !

% !5 5w

5

then

9 % 9 .Ð Ñ \ Ê \ Ð. ÑÞ Ð)Þ&%Ñ" (%% A

4Ä!

V

Now consider a general -net , not necessarily satisfying (8.53). Fix a. A -% %œ Ö ×4 4

partition with null boundaries, satisfying (8.44). If two partition elementsT œ ÖT ×! !5 5

overlap (e.g., over atoms), elements of are apportioned among them alternatingly.A%By (8.42),

R ÐT Ñ ´ l T l ŸG

% ! % !5 5A%

for some independent of and , while . LetG ! 5 R ÐT Ñ% % .% ! !5 5Ò% Ä !

E œ 5 À l R ÐT Ñ l Ÿ"

%! % ! ! Ÿ% .!

5 5

and

A% ! !ß

5−E

5œ T Þ Ð)Þ&&Ñ.%!

Note that as 0.A A %% !ß Å ÄWe claim that if , thenW § Ö4 À Â ×% %

% !- A4ß

115

9 %Ð Ñ \ Ê !Þ Ð)Þ&'Ñ"W

4Ä!

%

%%

The proof of this claim, briefly, follows by assuming the negation and forming analternative, as in (8.49). A contradiction then follows along similar lines.

Let , and! ! %œ Ð Ñ

A A‡ß ß Ð Ñ

Ÿ

% ! % ! %

% %

œ Þ,w

w w

Let ( ) 0 sufficiently slowly that as , and! % A A %Ò% Ä 0 ‡

ß% ! Å Ä !

9 %Ð Ñ \ Ê !ß Ð)Þ&(Ñ"W

4

%w

%

if ; this is possible by (8.56). For W § Ö4 À Â × T §% % !% % !w

4 5‡ ‡4 ß- A A

l R ÐT Ñ l Ÿ Þ"

% .!

% ! !5 5

Let be constructed so that if , andA - A A A A% %% ! % ! !% !" " "

4 4 5 5 5 ‡ßœ Ö × § T œ T T §

A A% ! % !"

5 5 T Tdiffers from by the minimal number of elements such that

¹ ¹% .!

R ÐT Ñ Ÿ Ð)Þ&)Ñ"

% ! !"

5 5

for all . Thus by (8.54),5

9 % - 9 .Ð Ñ \Ð Ñ Ê \ Ð. ÑÞ Ð)Þ&*Ñ" (4

4"

Ä!%

% AV

However, is a small perturbation of in that (8.57) and (8.59) imply that (8.59)A A% %" ß

holds also if is replaced by . - -% %4"

4 è

116

Chapter 9

Joint Distributions and Applications

We now use the mathematical machinery developed in the last two chapters toconsider joint distributions of integrals of r.v.'s and apply them to statistical mechanics.The work in deriving joint distributions of quantum observables will, with the aboveformalism, be minimal.

§9.1 Integrals of Jointly Distributed R.V.'sAs in Chapter 8, let be a metric space with a -finite Borel measure .Ð ß ÑA 5 5 .

Assume that random vector-valued functions have a jointXÐ Ñ œ Ð\ Ð Ñßá ß\ Ð ÑÑ- - -" 8

distribution for fixed and are independent for different . Let be- A - 9 ‘ ‘− À Ä

given. Define the direct product and for W W / / /8 8 " # 83œ" 3 3 3 3œ ‚ œ Ð ß ßá ß Ñß/

3 œ "ß #ßá ß let

3 3 / /9 98

" #

4œ"

8

" #4 4Ð ß Ñ œ Ð ß Ñ/ / "

(see equation (7.21)). Define

e . / %essÐ Ñ œ Ö − VÐ Ñ À Ð ÐF Ð ÑÑÑ ! a !×ÞX X X/ "%

Let ( denote the joint ch.f. of .FX t XÑ

Theorem 9.1: The random vector exists if and only ifY Xœ Ð Ñ Ð. Ð ÑÑ'A - 9 . -

Ð3Ñ − Ð Ñß for / eess X

< 9 $$

X XÐ Ñ ´ Ð? Ð Ð Ñ Ñ "Ñ Ð*Þ"Ñ"

t tlim$Ä!

exists, and

117

Ð33Ñ

< < . -Ð Ñ ´ Ð Ñ. Ð Ñ Ð*Þ#Ñt t(A

-XÐ Ñ

converges absolutely. In this case, the ch.f. of is Y / Þ<Ð Ñt

Proof: Suppose exists. If is an -vector and is the log ch.f. of , then forY a t Y8 Ð Ñ<> − Ð> Ñ †‘ <, is the log ch.f. of . By Theorem 8.5 the log ch.f. ofa a Ya Y a X† œ † Ð Ñ Ð. Ð ÑÑ Ð>Ñ . Ð Ñ' '

A A -- 9 . - < . - is also given by , adopting the notation ofa X† Ð Ñ

(9.1) for scalar r.v.'s. For a given , if then - A e - e− − Ð\Ñß † Ð Ñ − Ð † ÑÞX a X a Xess essThus, since F Fa X X† Ð>Ñ œ Ð> Ñßa

lim lim$ $Ä! Ä!

†" "Ð Ð Ð Ñ>Ñ "Ñ œ Ð Ð Ð Ñ> Ñ "Ñ$ $F 9 $ F 9 $a X X a

exists; since is arbitrary, follows. In addition, by Theorem 8.5,a Ð3Ñ

< < . -Ð> Ñ œ Ð>Ñ . Ð Ñßa (A

-a X† Ð Ñ

and the latter converges absolutely. Using we conclude (9.2)< <a X X† Ð>Ñ œ Ð> Ñaconverges absolutely, and has log ch.f. proving .Y t<Ð Ñß Ð33Ñ

Conversely, assume and hold. Then given and thereÐ3Ñ Ð33Ñ \ − Ð † Ñßa a Xw eessexists such that and . Thus by (9.1) andX X X a Xw w w w w w

" 8œ Ð\ ßá ß\ Ñ − Ð Ñ † œ \eess(9.2) for and then by Theorem 8.5, exists. Since was arbitrary, the proof is<Xw a Y a†complete. è

§9.2 Abelian -algebras[‡

We present here a capsule summary of the spectral theory of -algebras to be used[‡

in the next section. The material may be omitted without loss of continuity by thosefamiliar with it.

Let be a separable complex Hilbert space and be the bounded linear[ _ [Ð Ñoperators on . A -algebra on is an algebra of bounded operators on closed[ T [ [[‡

under adjunction , and closed in the weak operator topology on . A -ÐE Ä E Ñ [‡ ‡[algebra is naturally a normed linear space with norm inherited from Let be the_ [ TÐ ÑÞ ‡

space of bounded linear functionals on as a Banach space and let the spectrum of T TWconsist of those which are also multiplicative, i.e., , in9 T 9 9 9− ÐE E Ñ œ ÐE Ñ ÐE ч

" # " #

the weak-star topology inherited from The Gelfand representation gives a canonicalT‡Þisometric algebraic star-isomorphism of with the algebra (under multiplication) formedTby the bounded continuous complex-valued functions on . This isomorphism is,G ÐWÑ WF

for + − ßT

118

+ Ç 0 − G ÐWÑß 0Ð Ñ œ Ð+ÑÞF where 9 9

For let be the corresponding operator. For the map0 − G ÐWÑ X − Bß C − ßF 0 T [0 Ä ÐX Bß CÑ G ÐWÑ0 F is non-negative and is bounded on in its uniform (sup-norm)topology; hence there exists a Borel measure on such that for .BßC FW 0 − G ÐWÑ

ÐX Bß CÑ œ 0. Þ Ð*Þ$Ñ0 BßCW

( .

Definition 9.2: The measure is a . A measure on is if for. .BßC spectral measure basicWany subset of to be locally -null, it is necessary and sufficient that it be locally -W . .BßB

null for every .B − [

Clearly any two basic measures are absolutely continuous with respect to each other.We have (see [D2])

Proposition 9.3: If is separable, then carries a -finite basic measure.[ 5W

If is basic, then . If is a measure space, then , as a. . A AG ÐWÑ œ P ÐWß Ñ P Ð ÑF_ _

[ P Рч #-algebra acting on , is the of . We then have (seeA Amultiplication algebra[D2]):

Proposition 9.4: Let be basic on . Then the Gelfand isomorphism is the unique. Wisometric star-isomorphism of the multiplication algebra onto P ÐWß Ñ Þ_ . T

If is a -algebra on , a possibly unbounded closed operator is withT [[ E‡ affiliatedT (T T T, or , if commutes with every unitary operator in the commutant of IfE E Þ Ew

is normal, then if and only if where , and is Borel-E E œ 0ÐE Ñß E − 0 À Ä(T T ‚ ‚" "

measurable.A -algebra is maximal abelian self-adjoint (masa) if it is properly contained in no[‡

other abelian -algebra. In physics, a maximal commuting set of observables (that is,[‡

its spectral projections) generates a masa algebra. We require:

Theorem 9.5 Two masa algebras are algebraically isomorphic if and only if they[Se1]:are unitarily equivalent.

Definition 9.6: A measure space is if every measurable set is a least upperlocalizablebound of sets with finite measure in the partial order of set inclusion. Two measurespaces have isomorphic measure rings if there exists an algebraic isomorphism betweentheir (Boolean) rings of measurable sets (modulo null sets). They are if theisomorphicisomorphism preserves measure.

Theorem 9.7 Two localizable measure spaces have isomorphic measure rings if[Se7]:and only if their multiplication algebras are algebraically star-isomorphic.

119

§9.3 Computations with Value Functions

Although joint distributions of non-commuting observables in quantum statisticalmechanics are difficult to describe (see [Se6] for the groundwork of such analysis), this isnot the case with commuting observables, which are amenable to a standard(commutative) probabilistic analysis.

Let be an abelian -algebra, and let and . We assumeT (T[ E   ! E Ð" Ÿ 3 Ÿ jч3 3

that represent globally conserved quantities in a canonical ensemble whose densityE3

operator is formally

3 œ Ð*Þ%Ñ/

/

†. Ð Ñ

† Ð Ñ

" >

" >

A

Atr

where , an -tuple of positive numbers, , and" " "œ Ð ßá ß Ñ j œ ÐE ßá ßE Ñ" j " jA. Ð Ñ œ Ð. ÐE Ñßá ß . ÐE ÑÑ F ßá ßF> > > > > >A " j F J " 8 ( may be either or ; see §1.2). Let be self-adjoint operators affiliated with , whose joint distribution in the canonicalTensemble of (9.4) is to be determined.

As with the single operators, must be interpreted as a limit of density operators with3discrete spectra. Proceeding in an analogous manner, let be the spectrum of . UnderA TBose statistics, for , let be the probability space where- A a ™ U c− Ð ß ß Ñ- -™

™ U ™ c Рц D Рцœ Ö!ß "ß # á D − ß ÐDÑ œ / Ð" / Ñ, }, is its power set and for ,™ -- " - "

A A

where . Under Fermi statistics,AÐ Ñ œ Ð ÐE Ñßá ß ÐE ÑÑ- - -" j

a U c- -œ ÐÖ! "×ß ß Ñßß Ö!ß"×

where

c c c- - -" -

Ð!Ñ œ à Ð"Ñ œ " Ð!ÑÞ Ð*Þ&Ñ"

" / † Ð ÑA

Thus represents possible particle numbers in state . Spectral multiplicities need nota --

be indicated here by duplication of spectral values, since they will be subsumed in aspectral measure . Let be the direct product space with product measure. a aœ #

--

c cœ #-

-.

Definition 9.8: The pair is the over at generalizedÐ ß Ña c canonical ensemble Ainverse temperature corresponding to the (generally formal) operator ." 3

Let be an r.v. on defined by is formally the number ofR R Ð D Ñ œ D à R- - - - -- A

a #w

w

−

particles in state . For let- " Ÿ 3 Ÿ 8

\ Ð Ñ ´ ÐF ÑR3 3- - -

be the r.v. on representing the “total amount” of observable in state .a > -. ÐF Ñ3

120

Our goal is to ascertain the joint distribution of ; thisÐ. ÐF Ñßá ß . ÐF ÑÑ ´ . Ð Ñ> > >" 8 Bis the distribution of a formal sum of over . In orderXÐ Ñ œ Ð\ Ð Ñßá ß\ Ð ÑÑ −- - - - A" 8

to study the distribution asymptotically, we first center LetXÐ ÑÞ-

X X X BÐ Ñ ´ Ð Ñ Ð Ð ÑÑ ´ Ð ÑR ß Ð*Þ'Ñ- - X - - -

where R ´ R ÐR Ñß œ ÐF ßá ßF ÑÞ- - -X B " 8

Let be a ( -finite) basic measure (see §9.2) on . We study the Lebesgue integral. 5 A'A -

" -XÐ ÑÐ. Ð ÑÑ R /- . -"# . In the Bose case the r.v. is geometric with parameter , and † Ð ÑA

in the Fermi case it is Bernoulli with parameter given by (9.5). Let be the ch.f. ofG-Ð>ÑR Ð Ñ Ð Ñ- -. If is the ch.f. of , thenF -t X

F G -- -Ð Ñ œ Ð † Ð ÑÑÞ Ð*Þ(Ñt t B

Therefore, for small9

F 9 9 9Ð Ñ œ " SÐ Ñß"

#t tMt# w $

where is the covariance matrix of ,Q Ð Ñ œ Ð!Ñ34 -`` `

#

> >3 4

.- X

Q Ð Ñ œ ÐF Ð ÑF Ð ÑR Ñ œ F Ð ÑF Ð Ñ Ð*Þ)Ñ/

/ … "34 3 4 3 4

† Ð Ñ

† Ð Ñ#- X - - - --

" -

" -

#

A

AŠ ‹(see equation (2.10)). The holds for Bose (Fermi) statistics. Thus if , Ð Ñ Ð Ñ œ9 $ $

"#

< F $ -- -$

Ð Ñ ´ Ð Ñ " œ Ð Ñ Þ"

#t t tM tlim

Ä!

wŠ ‹"#

Together with Theorem 9.1 this proves the reverse direction of

Theorem 9.9: If are the centered random vectors above, then the LebesgueXÐ Ñ-integral

Y Xœ Ð ÑÐ. Ð ÑÑP

"#(

A

- . - Ð*Þ*Ñ

exists if and only if

M M´ Ð Ñ. Ð Ñ(A

- . - Ð*Þ"!Ñ

converges absolutely component-wise. In this case is normal, with covariance matrixY M.

121

Proof: Only the forward direction remains. If (9.9) exists, then by Theorem 9.1,

(A $

-limÄ!

"Ð Ð Ñ "Ñ . Ð Ñ

$F $ . -

"# t

converges absolutely for all . Thus, the same is true for (9.10). t è

§9.4 Joint Asymptotic Distributions By analogy with single operators, calculations like those above can be viewed as a

direct treatment of an infinite volume limit. However, the algebra and its spectrum T Aare actually limits of a net of discrete algebras with spectra , and the jointT A% %

distributions of are asymptotic forms of those for discretized random vectors . In theX X%

discretized situation, all formal quantities, including the density operator, are well-defined.

Let be a -finite basic measure on , and be a metric on whose Borel sets . 5 A 5 A Uare just the -measurable sets (the existence of such metrics in applications will be.shown explicitly; the integral, if it exists, is independent of the metric used). Let beÖ ×A%

a -net (Definition 8.9) of points in , and for , let .. A TE − E œ E%A¹

%

Definition 9.10: The algebra is the - of T T % T% %œ ÖE À E − × discrete approximationwith respect to . Let and be as in §9.3.. E ßá ßE F ßá ßF" j " 8

In calculating -integrals of r.v.'s we will be calculating V Ð Ñ œ Ð ÑR Ä !‡ X B- - %-

limits of sums

" "- A - A

% % %

% % % %3 3− −

3 3X XÐ Ñ ´ Ð ÑÞ Ð*Þ""Ñ- -

In order to use the machinery of §8.3, we need some assumptions about and .A BÐ Ñ Ð Ñ- -Precisely, we require that be -continuous, and similarly for (that is, thatE Ð Ñ F Ð Ñ3 3- 5 -there exist continuous representatives). We then have

Theorem 9.11: Let and be as above, and be uniform (DefinitionA BÐ Ñß Ð Ñß ß ß- - A . 5 .8.11) with respect to . If for some 5 % !

( Š ‹A 5 - - %

" -

" -

supÐ ÑŸ

#† Ð Ñ

† Ð Ñ#

"ß

l Ð Ñl . Ð Ñ _ß Ð*Þ"#Ñ/

/ … "B - . -

A

A

then the normalized distribution of in the canonical ensemble over at generalizedB Ainverse temperature is jointly normal, with covariance"

Q œ F Ð ÑF Ð Ñ . Ð Ñß Ð*Þ"$Ñ/

/ … "34 3 4

† Ð Ñ

† Ð Ñ#( Š ‹A

" -

" -

- - . -A

A

122

where refers to Bose (Fermi) statistics. That is, Ð Ñ

V‡(A

BÐ ÑR Ð. Ð ÑÑ- . --"Î#

exists and is normal with covariance .Q34

Proof: By Theorems 8.12, 8.7, and 9.9, we need only show that isX BÐ Ñ œ Ð ÑR- - -

Riemann intregrable. By Corollary 8.8.1, it suffices to show

7 Ð Ñ ´ Ð \ Ð ÑÑ Ð \ Ð ÑÑ − P Ð Ñ Ð*Þ"%Ñ"

% $ß3 "

"" 3 " " 3- X ) $ - ; $ - .

$sup$ $

5 - - %"

"

ŸÐ ß ÑŸ

Š ‹È È

for some , and all , where% $ ! " Ÿ 3 Ÿ 8

;"ÐBÑ œBà B Ÿ "!à B "Þœ

Since ( , letting X -\ Ð ÑÑ ´ ! MÐBÑ œ Bß3

7 Ð Ñ Ÿ \ lÐM ÑÐ \ Ñl"

Ÿ - Ð \ Ñ œ -F Ð Ñ ÐR Ñ"

œ -F Ð Ñ/

/

% $

-

" -

" -

ß3

"" " " 33

#

"" 3 3

# #

3#

† Ð Ñ

† Ð Ñ

- X $ ; $$

$X $ - i

-

sup

sup

sup

$ $5 - - %

$ $5 - - %

5 - - %

"

"

"

"

"

ŸÐ ß ÑŸ

ŸÐ Ÿ ÑŸ

Ð ß ÑŸ

ˆ È ‹

A

A … "Þ

This shows that (9.14) is implied by (9.12) and completes the proof. è

§9.5 ApplicationsAn advantage of Einstein space over canonical spatially cut-off versions of

Minkowski space is its possession of the full conformal group of symmetries. Inparticular, the rotation group acts on Einstein space. Untractable (non trace-class)expressions involving generators of the conformal group in Minkowski space becometractable in Einstein space. For the purpose of evaluating joint distributions ofobservables Minkowski 4-space should be viewed as an infinite volume limit of Einsteinspace.

Theorem 9.11 allows evaluation of joint distributions in canonical ensembles (9.4),viewed as limits (according to a spectral measure ) of systems with discrete spectrum..This section provides two explicit calculations.

Non-vanishing chemical potentialÐ3Ñ

123

Let be a system with chemical potential , single particle Hamiltonian , andf . ! Eformal density operator

3 œ Ð*Þ"&Ñ/

/

†. Ð Ñ

†. Ð Ñ

" >

" >

A

Atr

where , and . Note that is the particle number operator." " . >œ Ð ß Ñ œ ÐEß MÑ R œ . ÐMÑAThis models an ensemble in which creation of particles requires energy In this case the.Þ[ E M‡-algebra generated by spectral projections of and is the bounded BorelTfunctions of .E

If consists of non-relativistic particles in Minkowski -space, the spectrum off 8 "T ‘ is measure-theoretically equivalent to . The appropriate spectral measure is

.7 œ I .I L œ . ÐEÑ R8 8"1

>

8Î#

8##Š ‹ (see (6.12)). The joint distribution of and is>

obtained by integrating , where and is a centeredX B Bœ ÐIÑR ÐIÑ œ ÐIß "Ñß RI I

geometric r.v. with parameter (under Bose statistics). According to Theorem/ I" .

9.11, the normalized joint asymptotic distribution of and is normal with covarianceL R

M œ .7ÞI II "

/

Ð/ … "ÑV‡( A

" .

" .

# I

I #

Defining the generalized zeta function

'Ð8ß BÑ ´ 5 B Ð*Þ"'Ñ"5œ"

_8 5

and using

(!

_ 8 B

B #B /

Ð/ … "Ñ.B œ „ 8x Ð8ß „ / Ñ Ð8   #Ñß Ð*Þ"(Ñ

.

..'

we get

M œ Ð8   $Ñ„ 8x Ð8 "ß „ / Ñ Ð8ß „ / Ñ

Ð8ß „ / Ñ Ð8 "ß „ / Ñ

1

" >

' '

' '

8# #

8 8##

8Ð8"Ñ 8

8 Š ‹ " ". .

". .

.

Note that and coincides with (6.25) and the right side of (6.21), respectively, inQ Q"" ##

the Bose case if . The calculation for is similar, and thus omitted.. œ ! 8 $

Density operator involving angular momentum (see [JKS])Ð33Ñ In this model, the formal density operator in a system is given byf

3 œ à Ð*Þ")Ñ/

/

L P

L P

" #

" #

#

#tr

124

where and are energy and angular momentum [JKS, Se8]; we ignore chemicalL P#

potential for simplicity. We will study in Minkowski four-space , as an infinitef Q%

volume limit of Einstein space.We remark first on the appropriate measure space of Theorem 9.11. Let be theT‡

[ E 7‡ #-algebra generated by and , the single particle energy and angular momentumoperators. Let be the spectrum of and be a -finite basic measure on ByA T . 5 A‡ ‡ ‡ ‡ß ÞProposition 9.4, is star-isomorphic to P Ð Ñ Þ_ ‡ ‡. T

Let , and , with Lebesgue measure, and ,A ‘ ™ .œ ‚ œ 7‚ - 7 -Ö6× œ #6 "

for . Then is star-isomorphic to , specifically through the6 − œ P Ð Ñ™ T . T _ ‡

correspondence

E Ç Q à 7 Ç Q Ð*Þ"*ÑI#

6Ð6"Ñ

where and denote independent variables on and , respectively, and denotesI 6 Q‘ ™

multiplication. Thus and are star-isomorphic. By Theorem 9.7,P Ð Ñ P Ð Ñ_ ‡ _. .therefore, and have isomorphic measure rings, so that the image (underÐ ß Ñ Ð ß ÑA . A .* ‡

the isomorphism) of on is equivalent to , and thus itself a basic measure.. A .‡ ‡

Thus, henceforth we may restrict attention to is a physically appropriateÐ ß ÑàA . .measure on , since it incorporates the asymptotics of the joint spectrum of and inA E 7% %

#

Einstein space of radius proportional to as becomes infinite. SpecificallyY V ß V% "%

(see §6.3), the joint spectrum of and in isE 7 Y% %# %

ÖÐ 8ß 6Ð6 "ÑÑ À 8ß 6 − ×ß% ™

this being the joint range of the corresponding functions on .AThe object of interest in studying the asymptotics of the joint distribution% Ä !

(covariance) of and is, according to Theorem 9.11, the local covariance of L P IR#-

and , where , and is a centered geometric r.v. with6Ð6 "ÑR œ ÐIß 6Ñ − R- -- A

parameter , with and . This is the dyadic matrix/ œ Ð ß Ñ Ð Ñ œ ÐIß 6Ð6 "ÑÑ † Ð Ñ" -A " " # -A

MÐ Ñ œ ÐI 6Ð6 "ÑÑ ÐR Ñ œ ÞI I I6Ð6 "Ñ

6Ð6 "Ñ I6Ð6 "Ñ 6 Ð6 "Ñ

/

/ … "Ñ- i -

"

"

#

# #

†

† #

A

A(

Integrating over , we obtain as the covariance of and in Minkowski spaceA L P#

M œ Ð#ß Ñß Ð*Þ#!Ñ"

"0 #

# " ``

" ` `` `

…" " #

" # #

#

#

#

where

0 # '…

6œ!

_ 6Ð6"ÑÐ8ß Ñ œ „ Ð#6 "Ñ Ð8ß „ / Ñ Ð*Þ#"Ñ" #

with given by (9.16). Note that, as indicated by the limit in (9.21), the joint' # Ä !distribution of energy and angular momentum when is singular, with the# œ !conditional expectation of angular momentum infinite for every energy value inMinkowski space. This is to be expected, since for a fixed energy value the range of the

125

angular momentum becomes unbounded as . See [JKS, Se8] for an application ofV Ä _(9.20) to a model for the influence of angular momentum on the cosmic backgroundradiation.

126

Epilogue

I would like to leave the reader by briefly identifying two significant open questionswhich arise in the present context.

The first is, what asymptotic probability distributions arise in a system whosespectral measure fails to have three continuous derivatives near 0. µ B .B Ð+ !Ñ. !

(Definition 3.11)? The failure to answer this question in Chapters 4 and 5 seemstechnical.

Second, how can these results be extended to a non-commutative setting (i.e., oneinvolving non-commuting observables)? It seems that gage spaces, the non-commutativeanalogs of probability spaces [Se6], are the appropriate framework. This question mayhave very significant mathematical ramifications.

127

References

[AS] Abramowitz, M. and I. Stegun, , U.S.Handbook of Mathematical Functions Printing Office, Washington, D.C. 1968Þ

[BD] Bjorken, J.D. and S.D. Drell, , McGraw-Hill,Relativistic Quantum Fields New York, 1965Þ

[BDK] Bretagnolle, J., D. Dacunha Castelle, and J. Krivine, Lois stables et espaces , 2 (1966), 231-259.P: Ann. Inst. H. Poincaré

[Ch] Chung, K.L., Academic Press, New York,A Course in Probability Theory, 1974.

[Che] Chentsov, N.N., Lévy-type Brownian motion for several parameters and generalized white noise, 2 (1957).Theor. Probability Appl.

[D1] Dixmier, J., - , North Holland, Amsterdam, 1977.G‡ Algebras

[D2] Dixmier, J., North Holland, Amsterdam, 1981.Von Neumann Algebras,

[GS] Gelfand, I.M. and I.S. Sargsjan, , A.M.S.,Introduction to Spectral Theory Providence, 1975.

[GV] Gelfand, I.M. and N.J. Vilenkin, , vol. 4. Generalized Functions Some Applications of Harmonic Analysis, Academic Press, New York, 1964.

[GV] Gnedenko, B.V. and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, 1968.

[GR] Gradshteyn, I.S. and M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965.

[G] Gumbel, E.J., Columbia University Press, New York,Statistics of Extremes, 1958.

[H] Hörmander, L., The spectral function of an elliptic operator, 121Acta Math. (1968), 193-218.

[KL] Khinchin, A.Y., and P. Lévy, Sur les lois stables, C.R. Acad. Sci. Paris 202 (1936), 374-376.

[JKS] Jakobsen, H., M. Kon, and I.E. Segal, Angular momentum of the cosmic background radiation, 42 (1979), 1788-1791.Phys. Rev. Letters

128

[Kh] Khinchin, A.Y., GraylockMathematical Foundations of Quantum Statistics, Press, Albany, N.Y., 1960.

[La] Lamperti, J., Semi-stable stochastic processes, Trans. Am. Math. Soc. 104 (1962), 62-78.

[LL] Landau, L.D. and E.M. Lifschitz, , Addison-Wesley,Statistical PhysicsReading,

Mass., 1958.

[Lé] Lévy, P., A special problem of Brownian motion and a general theory of Gaussian random functions, in Proceedings of the Third Berkeley Symposium on Mathematical and Statistical Probability, 1956.

[Lo] Loève, M., Springer-Verlag, New York, 1977.Probability Theory, I,

[Mc] McKean Jr., H.P., Brownian motion with several dimensional time, Teor. Verojatnost. i Primenen 8 (1963), 357-378.

[M] Malchan, G.M., Characterization of Gaussian fields with the Markov property, Soviet Math. Dokl. 12 (1971).

[R1] Rozanov, Ju. A., , Springer-Verlag, New York, 1982.Markov Random Fields

[R2] Rozanov, Ju. A., On Markovian fields and stochastic equations, Mat. Sb. 106(148) (1978), 106-116.

[See] Seeley, R., An estimate near the boundary for the spectral function of the Laplace operator, 102 (1980), 869-902.Am. J. Math.

[Se1] Segal, I.E., Decompositions of Operator Algebras: II: Multiplicity Theory, American Mathematical Society Memoirs 9 (1951), 1-66.

[Se2] Segal, I.E., AcademicMathematical Cosmology and Extragalactic Astronomy, Press, New York, 1976.

[Se3] Segal, I.E., Tensor algebras over Hilbert spaces. I., 81Trans. Am. Math. Soc. (1956), 106-134.

[Se4] Segal, I.E., Tensor algebras over Hilbert spaces. II. 2Ann. Math. (1956), 60-175.

[Se5] Segal, I.E. and R.A. Kunze, , McGraw-Hill, New York,Integrals and Operators 1968.[Se6] Segal, I.E., A noncommutative extension of abstract integration, 57Ann. Math.

129

(1953), 401-457.

[Se7] Segal, I.E., Equivalences of measure spaces, 73 (1951),Amer. J. Math. 275-313.

[Se8] Segal, I.E., Radiation in the Einstein universe and the cosmic background, 28 (1983), 2393-2401.Phys. Rev. D

[Si] Simon, B., Schrödinger semigroups, 7 (1982), 445-526.Bull. Am. Math. Soc.

[V] Vanmarcke, E., R , M.I.T.andom Fields: Analysis and Synthesis Press, Cambridge, U.S., 1983.

[WW] Whittaker, E.T. and G.N. Watson, CambridgeA Course of Modern Analysis, University Press, London, 1973.

130

Index

almost surely (a.s.), 6, 38angular momentum, 1, 4, 123-125 covariance with energy, 124antisymmetric distributions, 32 occuptation numbers, 32 statistics, 8, 32antisymmetrized tensor product, 8asymptotic distributions, under symmetric statistics, 27, 47, 64, 65-68, 71-74 under antisymmetric statistics 32, 33asymptotic energy density, 67, 68, 70, 74asymptotic energy distributions, 67asymptotic probability distributions, 126asymptotic spectrum, 15, 16Banach space, 117basic measure, 118, 124Bernoulli, 14, 32, 120Bessel function, modified, 64blackbody curve (classical), 60Boltzmann's constant, 4, 13Borel measures, 83 (finite), 86, 95-6, 111 (metric space), 118 (measurable) positive, 86 set 83, 94, 121 Borel measurable, 9639 -finite, 110, 1165Bose-Fermi ensembles, 6, 74 statistics 74, 119, 120, 122Bose condensation, 74Boson field, 7 energy, 70 number, 72-3bounded functions, 123bounded r.v.-valued function, 85, 95bounded variation, 18, 27, 28Brownian motion, 76Cauchy data, 63, 64characteristic function (ch.f.), 6, 54, 60, 61 , 106, 116ß *"canonical ensemble (over an operator), 11, 13, 17, 119, 121central limit theorem, 23chemical potential, 6, 60, 123, 124 non-zero, 123

131

non-vanishing, 122convergence in law, 6, 24, 42, 86convergence distributional, 38 order-independently, 78, 87 weakly, 77-8, 80-1, 100, 101convolution, iterated, 23, 53 (exponential)countably additive measure-valued measure, 103distribution function (d.f.), 6, 78 (cumulative)density of states, 60, 61density operator, 4, 11, 64, 119, 123density (mean) of photons, 74dimensions (of a set), 111dimensions, physical, 6Dirac delta distribution, 63discretized Gaussian, 16dominated convergence theorem, 110dyadic matrix, 124eigenvalue density, 17Einstein space, 4, 5, 60, 69, 71, 75, 122, 124 distributions in 2 dimensions, 124 distributions in 4 dimensions, 70-75 comparison with Minkowski space, 122elliptic operator, 4, 16 (differential)energy density spectrum, 5energy distribution function, 67%-discrete approximation, 121%-discrete operator, 17, 18, 29, 61, 64, 71%-net of functions, 17equilibrium quantum system, 4Euler's constant, 47even partition, 21, 22extreme value distribution, 47, 60Fermion field, 7 number 68, 69 (ensemble), 73, 74Fourier transform, 51, 63Fubini theorem, 110Gelfand isomorphism, 118Gelfand representation, 117Hamiltonian, 4, 5, 9, 11, 13, 20, 60, 64, 69, 70, 74, 123Hilbert space, 7, 9, 39, 62, 70 complex, 117Huygens' principle, 69hyperbolic equation, 63infinite volume limit, 60infinitively divisible distributions, 94

132

infinitesimal array of r.v.'s, 77 net of partitions, 85 sequence of partitions, 86infrared catastrophe, 34, 74integral of r.v.'s, 20, 76, 94, 116 of measure-valued functions, 71, 83integration by parts, 101integration operator, 94inverse temperature, 4joint characteristic function, 14joint distributions, 119-122, 123, 124joint spectrum, 124Khinchin, A.Y., 5Laplace-Beltrami operator, 69Lebesgue integral, 5, 7, 76, 83-89, 90 (def.), 91, 120 additivity, 87 existence, 96, 100, 103, 104, 120 finite, 85, 86 integrable, 108 integration, 76 general, 85, 102 measure, 124 of r.v.-valued function, 95 of vector functions, 116 -Lebesgue integrability, 959 relation to Riemann integral, 106, 107 sums, 87Lévy-Cramér convergence theorem, 37, 51Lévy-Khinchin transform, 77, 78, 80, 96, 104Liapounov's theorem, 24, 25limit theorems (general), 77limiting distribution, 23localizable (measure space), 118Loève, M., 77, 78, 105log ch.f., 94Lorentz, 63many particle state, 11maximal abelian self-adjoint (masa), 118mean value theorem, 79, 82, 109measure ring isomorphisms, 118measure space, 118metric Lévy, 83 , 8439metric space, 111

133

monotone decreasing, 36 non-decreasing, 41, 43multidimensional white noise, 5multiplication algebra, 118Minkowski space, 4, 5, 60, 62, 67-9 ( -dimensional), 74 ( space)8 " 8 " 75, 122, 123 ( space), 124, 1258 "net of operators, 4, 61 (discrete)non-atomic, 85, 112non-commutative probabilities, 126non-degenerate interval, 21non-increasing function, 18non-negative, 6, 10non-normal distributions, 6non-vanishing densities near zero, 51number random variable, 14,15occupation number, 13, 14, 29, 41 (total), 55, 61one parameter family of r.v.'s, 20order-independent, 79orthonormal eigenbasis, 13 basis, 9, 109-bounded, 85particle number operator, 123partition, 20, 21 disjointly, 86 even, 21photons and neutrinos, 60, 74-5Planck law, 4, 5, 6, 60, 74 constant), 75point mass, 87positive, 6, 20 (numbers)positive reals, 93probability distributions (of observables), 1, 23, 87-8, 94pure point spectrum, 11, 13quantization (of an operator), 9V‡integral, 110 definition, 110, 111 relation to Riemann integral, 111random distributions, 5random variable (r.v.), 6, 23, 34, 47 (geometric), 64random variable on canonical ensemble, 14random variable-valued functions, 5, 6, 94-5, 103, 106random variables with infinite variance (sums of), 77random vectors, 116range, essential, 85Riemann integral, 5, 20 (Riemann-Stieltjes), 76, 106, 107 ( -Riemann)9 definition, 106, 107 geometries, 60

134

integration, 76 integrable, 108, 112, 114, 122 manifolds, 65, 76 partial Riemann sum, 114 relation to Lebesgue integral, 94, 107 relation to -integral, 111V‡

zeta function, 65Riemannian manifold, 4, 16scaling stable, 88, 89, 91stable (scaling) distribution, 88-90Schrödinger operator, 60, 61Segal, I.E., 4, 6self-adjoint, 9, 10self-adjoint operator, 16semi-stable process, 91semi-stable stationary 0-mean, 92separable, - , 959sequence of partitions, 225-finite basic measure, 124 measure space, 85single particle energy, 124single particle space, construction, 63-65single particle state, 11singletons, 89singular integrals, 26, 34-36spectral approximation, 61, 64, 65spectral density (non-vanishing) 5, 16spectral function, 17, 29, 38, 43, 57, 61spectral measure, 4, 17, 18, 19, 20, 27, 39, 47 (with non-vanishing density), 60, 61, 64, 65, 71, 118, 123, 126spectral -net, 20, 21, 23, 25, 27.stable distribution, 23, 24, 88 characterization, 88, 89 star -isomorphic, 124standard deviation, 23state probability space, 10, 11Stieltjes integral, 17, 20, 31Sturm-Liouville systems, 16symmetric canonic ensemble, 11, 29symmetric occupation number r.v.'s, 39symmetrized tensor product, 7tail behavior of probability distributions, 83tails of integrated distributions, 77topology of weak convergence, 77trace class, 4, 10, 11, 12, 18, 19 under antisymmetric statistics, 11

135

uniformly bounded, 82value function, 5, 6, 10, 11, 13, 29von Neumann algebra, 76 (abelian), 110[‡-algebras (abelian), 117 definitions, 117, 124 properties, 117, 118without loss of generality (w.l.o.g.), 7, 34, 81, 87, 92, 990-free (operator), 11, 19, 270-freedom condition, 320-free spectral measure, 190-mean random variables, 23

136

List of Symbols

Page

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6 G_-

™

‘

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Z Ð Ñ8ÐFÑ

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O8ÐJÑ

Z Ð Ñ8ÐJÑ

5

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^F

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10 T4

n as

aµ

^ K G11 n4

12 OF

OJ

13 . ÐGÑ> "

5"

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16 cÐ8 œ 6Ñ c ? %Ð œ 6Ñ17 JÐIÑ E%

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,

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21 .ÐT Ñ8 .ÐI Ñ4 .ÐI Ñ3 Ö I ×? 83 3 M% 8

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