The gamma process and the Poisson distribution

Steutel, F.W.; Thiemann, J.G.F.

Published: 01/01/1989

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Citation for published version (APA):Steutel, F. W., & Thiemann, J. G. F. (1989). The gamma process and the Poisson distribution. (MemorandumCOSOR; Vol. 8924). Eindhoven: Technische Universiteit Eindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum COSOR 89-24

The gamma process and the

Poisson distribution

F.W. Steutel and J.G.F. Thiemann

Eindhoven University of Technology

Department of Mathematics and Computing Science

P.O. Box 513

5600 MB Eindhoven

The Netherlands

Eindhoven,September1989

The Netherlands

THE GAMMA PROCESS AND THE POISSON DISTRIBUTION

F.W. Steutel and J.G.F. Thiemann

1. INTRODUCTION

This paper developed from the following simple problem. Let K 1(JJ.), ... ,Kn(JJ.) be Li.d.

having a Poisson distribution with mean IJ.. Let K l;n ~ •.. ~ Kn;n denote the corresponding

order statistics. Problem: detennine EKj;n and var Kj;n'

It turns out that for n = 2 the answer can be given explicitly; we give the solution for K2;2

(see [7] for details):

EK2;2 =IJ. + jle-21l(/o(2lJ.) + 11 (2J.L» ,

var K 2;2 =J.L - J.L2e-411(1 0 (2J.L) + 11 (2J.L)i + jle -2jLI o(2J.L) ,

where I j denotes a modified Bessel function of order j.

For n > 2 the problem seems intractable, so we look for approximations for large J.L.

When approximating the Poisson distribution by a nonnal one, there is always the problem

of continuity corrections. We try to avoid that by first considering a continuous variant of the

Poisson distribution, and that is where the Gamma process comes in.

Let Z (t), t > 0, be a Gamma process with unit mean, i.e. Z is a process with stationary

independent increments and Z (1) has an exponential distribution with mean one. Now let T (JJ.) be

the exceedance time oflevellJ.. i.e.

(1) T(JJ.) =inf {t > 0 I Z (t) > IJ.}

Then

(2) (T(JJ.)~ t} ={Z(t) > Il} a.s.

and so

00 1-1 _%

(3) P(T(JJ.)~ t) =J x r(~ dx (t > 0) .

Now T(JJ.) has a continuous distribution function which is increasing on (0,00). Moreover, for

- 2 -

integer arguments it coincides with the Poisson distribution function with mean Il, since from (3)

we easily deduce

k-l :(4) P(T{Jl)~ k)= L III1 e-j.l (k =1,2, ... ) .

1=<J •

For future reference we introduce some notation. For each real number x let [x] be its integer

part, Le. the largest integer not exceeding x, and put {x} :=x - (x], the fractional part of x. The

property ofT mentioned above can then be states as: [T{Jl)] has a Poisson distribution with mean

J.L As a result, this paper contains some new results for exceedance times in Gamma processes

and an approximate solution of the above-mentioned problem about order statistics.

Before discussing the special case of the Gamma process we consider exceedance times of

constant levels for more general processes.

2. EXCEEDANCE TIMES IN PROCESSES WITH STATIONARY, INDEPENDENTINCREMENTS

In this section Z(t), t > 0, is a non-negative process with stationary, independent incre­

ments, scaled such that Z (0) =0 and EZ (1) =1. For each Il > 0 we define the exceedance time

T{Jl) by (1), as before and we shall be interested in the behaviour of [T{Jl)] and {T{Jl)} for large

values ofll (see Fig. 1, which is slightly misleading since the paths ofZ are not continuous)

Z(t)

O~---------=T~{Jl--:')--------t

Figure 1

The random variable T (Jl) is a.s. finite; per (Jl) > n) ~ P (Z (n) ~ Il) -+ 0 (n -+ 00).

Since the process Z is continuous in probability and non-decreasing, it is at each point a.s.

continuous. From this one can deduce that, for each t > 0 and for all but at most countably many

Il> 0, the sets

(Il < Z(t)}, (Il~ Z(t)}, {T{Jl) < t} and {T{Jl) $ t}

are a.s. equal.

- 3 -

We are now ready to prove the following result.

Theorem 1. When Z (l) is non-lanice, then

(5) lim P({TijL)} < u) =u (0 < u < 1) .Il-+OO

Proof: For each u E (0, 1) we have a.s.

-({TijL)} < u} = U {k~ T(~) < k +u} =k=O

-=U {Z(k)~ ~ < Z(k + u)}k=O

So_ Il

P({TijL)} <u)= 1: JP(Z(U»Il- X)dFZ (k)(X) ,k=OO

where Fz denotes the distribution function of Z. An appeal to the key renewal theorem (see [4])

easily yields

lim P({T<lL)} < u)= EZ(u) =uIl-- EZ(I)

o

dRemark 1. If one writes Z (l) =Z (l - u) +Z (u) then Theorem 1 is a special case of a well-

known result for alternating renewal processes.

Remark 2. Theorem 1 also holds without the non-lattice condition. The proof for the lattice case

can be given along similar lines.

Laplace transformation with respect to ~ is an efficient tool for obtaining asymptotic results

for ~ ----+ 00. In view of this, as a preparation for the special case in the next section we give a few

lemmas. For this purpose we need some more notation. Since Z(l) has an infinitely divisible dis­

tribution we have, for t, s > 0,

Ee-sZ{l) =,(S)' =e-hll{S) ,

with

(6) cXs):=Ee-sZ{l) and 'V(s)=-Iog,(s) .

We shall use the following simple fact: if F is a distribution function on [0. 00), then

- -- 4-

(7) i e-u dF(x) =s i e-SXF(x) dx (s > 0) .

Lemma!.

-I e-SIlFT(jl.)(t)d~=s-l(l-e-t'l'(S» (s, t>O).o

a.s.Proof Since {Il < Z(t)} c {T{JJ.)$ t} c {Il$ Z(t)}, we have, for all but at most countably many

Il"S,FT(jl.)(t)= I-Fz(l) (JJ.). So

- -I e-.fll FT(jl.)(t) d~= J e-SIl(l- FZ(t) (JJ.» d~ =o 0

-= s-1 - S-1 J e-.rlldFz(t)(~) =S-1 - s-1 Ee-sZ(I) =

o

o

Lemma 2.

-I e-sIlEe-'tT(jl.)d~ = \jf(S) (s, 't > 0) .o S ('t + 'I'(s»

Proof: By (7) we have

-Ee-'tT(jl.) = 't J e-'tt FT(jl.)(t) dt .

o

Laplace transfonnation with respect to Jl., an appeal to Lemma 1 and a change of order of integra­

tion, give the required result. 0

Lemma 3.

-OJ e-sll E (T (JJ.)k) dJl. = k! (s > 0, k E IN) .

S'I'(s)k

- 5 -

Proof: We get the result by differentiating k times both sides of the equality in Lemma 2 and let­

ting t .1 o. 0

Since TUJ.) is non-decreasing in 1.1., as follows from the definition of T, Lemma 3 implies that TUJ.)

has finite moments of all orders.

Lemma 4. For each u e (0, 1) the function 1.1. ~ F (T(1l)} (u) is the difference of two increasing

functions. and

-J -.fJl F ( ) d - 1~(s)W ( 0)

o e IT{Jl)) u 1.1. - s (1 _ C\l(s» s > .

Proof: Let u e (0,1). Then, for all but at most countably many I.I.'S, by (6) we have

-F IT{Jl))(U) = 1: P(k ~ T(I.I.)~ k + u) =k"()

-=1: [FT{Jl)(k + u) - FT{jL)(k)]k..Q

Hence. by Lemma 1.

-f e-SJl F (T{jL))(U) dl.l. = ~ [S-l(l- e-(k+w}1v(s» - s-l (1 - e-k1v(s»] =o k..Q

= 1 - e-IV(s) _ 1 - cHs)W

s(l-e4ll(s» - s(l-C\l(s»

Furthermore, for all 1.1. > 0, we have_ _ 00

FITM](U) = 1: P(k~ TUJ.)~ k+u)= 1: P(TUJ.)~ k)- 1: P(TUJ.) > k+u) ,k"() k"() k"()

where convergence of both series follows from majorization by the integral-f P (TUJ.) > x) ax = EI'UJ.) < 00. So F (T{Jl)] (u) can be written as the difference of two increasingofunctions of j.I.. 0

In the Lemmas 1-4 we have Laplace transforms of functions that are monotone or equal to

the difference of two monotone functions and, hence, the inversion theorem for the Laplace

transformation is in force (see e.g. [10] §7.3). When the random variables Z(t) (t > 0) have con­

tinuous distributions then these functions are continuous and therefore they can be obtained by

application of the inversion theorem to their Laplace transforms. We use Lemmas 1,3 and 4 to

obtain asymptotic results as 1.1.~ 00.

·6-

Theorem 2.

i)

ii)

lim E (T(jJl)/ JJ.k =1 (k =1,2, ... )Il-+OO

M

lim M-1 JF IT(jl») (u) dJJ. = u (0 < u < 1)M-+oo 0

Proof: We apply a Tauberian theorem (see Feller, vol. 2 Th. XlII.S.1) to the Laplace transforms

in the Lemmas 1,3 and 4.

For the function FT(jl)(l) we have

M 00

lim JFT(jl)(l) dJj. = JP (JJ. < Z (1» dJj. =EZ (1) =1 ,M-- O 0

So for its Laplace transform, given by Lemma I, we have, by the Tauberian theorem,

lim s-l(l-e-ll/(I»= I, which implies lim S-l'l'(S) = 1..r.j. 0 s.j. 0

Next we consider the Laplace transform f (s) of E (T (JJ.)k). By Lemma 3 and the result just

obtained we have lim sk+lf (s) =lim k !Sk'lf{s)-k =k!. From this i) follows by the Tauberians.l.o 1.1.0

theorem.

Finally, for the Laplace transform g(s) of F (T(jl»)(U) by Lemma 4 we have lim sg(s) =u,s.l.o

which implies ii) by the Tauberian theorem. 0

To obtain Theorem I, which is stronger than ii), a more powerful Tauberian theorem would

be needed. For the special case of the Gamma process much sharper results will be obtained from

the Lemmas 3 and 4.

3. THE GAMMA PROCESS

This is the particular case we are most interested in. Now we have (cf. (6»

1cKs) =-1- and 'I'(s) =log (1 + s) .

+s

Moreover, the random variables Z (t) (t > 0) have continuous distributions, so the functions

occurring in the Lemmas 3 and 4 can be obtained by applying the inversion formula to their

Laplace transforms. This will give us quite sharp versions of the Theorems 1 and 2.

- 7 -

Theorem 1'.

0< F (T(p.)}(U) - U < (1tl.1elLrl (0 < u < 1, 0 < jl) .

Proof: Let U e (0, I). For the Laplace transform fof F {T(p.)J (u) we have, by Lemma 4,

f (s) = 1- cp(s)U =s-2[1 + s - (l + s)l-u] .s(l-cjl(s»

Therefore f (s) =0 ( IS 1-1) (I s I~ 00), f has a pole in 0 and a branch point in -1. Conse­

quently, in the version fonnula

1 a+ib

F{T(p.)}(u)=-2. lim J e/lSf(s)ds1tl b~ a-ib

we can modify the path of integration as shown in Fig. 2. Since the residue of the integrand in 0

equals U, we get

-I o

~IIIIII

a

s-plane

Figure 2

-1

F (T(p.)}(U) =U - _1_. J ellSs-211 + s 11- ue -1l:i(l-u)ds +21tl -00

- 8 -

1 -1= U+ - J ellSs-Zll +s 11-

usin x(l-u) dsX __

So

1 -1 1O<F{TijL)}(U)-u<- J ellSds=-e-Jt

x __ nil

o

Theorem 2'. For all k e IV and Il > O.

"IE (T{IJ.)")/k! - ~ C-IIiIi! I ~ (Jt"+llJ.e l1r 1•

I=IJ

where the coefficients C-l are defined by

00

[Iog(l+s)r"= ~ CIS1 (lsi <1).

1-"

Proof: Let ke IV. By Lemma 3 the Laplace transform of E(T{IJ.)")/k! is s-l[1og(l+s)r".Exactly the same procedure as used in the proof of Theorem 2' yields

"E (T ijL)") =~ C-l~lll! +1=0

-1

+ 21

. I e llSs-1[log 11 +s I -xi]-~ds +Xl __

--+ 2

1. f e llSs-1[10g 11 +s I -xir"ds •

Xl -1

where again the first term on the right is the residu in O.

Now both integrals are easily seen to be bounded in absolute value by1 -1

2I efJ.Sn-"ds = t (Jt"+llJ.e l1r 1

• hence the result.1t __ I]

The following corollaries are important for our purposes. Corollary 2 is rather surprising andshows a behaviour that is similar to that observed in [6] for Y. {YIe} and [YIe]. {Y Ie} as e J, o.We recall that [TijL)] has a Poisson distribution with mean).1. and we refer to Theorem l' as wellas to Theorem 2'.

- 9-

Corollary 1.

ETijL)=J.l+t + o(e 41) ,

varT(J.L)=J.l- 112

+ o(e 41) ijL..-+oo) .

Corollary 2.

cov(TijL), (TijL)} = 0 (e-IL ) ,

1cov([T ijL)], (TijL)})=-U+ O (e 41) ijL..-+oo) .

From Corollary 2 it follows that in

d(8) KijL) =[TijL)] =TijL) - (T{J.l)} ,

the random variables in the right-hand side are practically uncorrelated (for an interesting casewhere [X] and {X} are independent we refer to [9]). In combination with Theorem l' it followsthat, as far as the first two moments are concerned, K ijL) is quite well approximated by

K (J.l) :: TijL) - U ,

where TijL) and U are independent and U is uniformly distributed on (0, 1).

4. APPROXIMATING Kj;lIijL)

In this section we derive approximations for EKj ;lIijL) and var Kj;lIijL) for large J.l. Startingfrom (8) and using a normal approximation for Tj;lIijL).

Let Il > 0. Since the distribution function F of T ijL) is continuous and increasing one has

dTijL)=F-1(U) ,

for any random variable U that is uniformly distributed on (0, 1). In particular we have

dT ijL) = F-1C1>(X) ,

where X is a standard normal random variable and C1> its distribution function. The right-hand sideof this equality depends on Il via F, and we make this dependence explicit by writing

d(9) T{J.l) =G(X, J.l) .

As both F and C1> are increasing functions, G is increasing in its first argument and therefore therelation (9) implies a similar relation for the order statistics corresponding to TijL) and X:

- 10-

d

(10) Tj;,,(JJ.) = G (Xj;"d.1) .

For the function G we have the following result, which we give without its straight-f01ward but

rather lengthy proof.

Lemma S. Let q be defined by

(11)

Then for G as defined by (9) one has

G (x, J.l) = q (x, J.l) + r (x, J.l)

with

1x4+1

I r (x, ~) I ~ C -- for Ix I ~ J.l 6J.l

and J.l sufficiently large, where C is a constant.

The expansion in (11) is very similar to an expansion given by Riordan [8J without error term,

and is related to Edgeworth expansions.

It is helpful for the intuition to combine (10) and (11) to

1

on {Xj;" ~ J.l 6 }.

In order to relate the expectation and variance of Tj;,,(JJ.) to those of Xj;" we need one

further estimate on G (x, ~), which enables us to obtain bounds on the tails of the corresponding

integrals.

1

Lemma 6. For x and J.l such that 2~ J.l 6 ~ Ix I we have

G (x, J.l)~ X 14er/(2jl,)

q (x, J.l) S 3x6 •

Proof: Let t be such that FT{jL)(t) = ~(x). We shall prove that t ~ 2~21x Ier/(4i) and so we may

suppose 2J.l2 < t.

- 11 -

Now let k e IV be such that

~2 _ 1S k S ~ < k + 1

then

112k ~2k ~2 k ~2 1i..S ...l::.....- < - (-) < (_)1& < ( )It

(2k)! - (k+1)k - k+1 - k+1 - t

and

hence

1

Since t = G (x, ~), from (13) and 2S ~ 6 S Ix I the estimates for G and q are easily deduced. 0

Lemmas 5 and 6 imply

Theorem 3.

d

Finally, we need to go from the random variable Tj;II{J.L) to [Tj ;II{J.L)] =Kj;II{J.L). This is done in thefollowing lemma.

Lemma 7. For each j, n E IN with j S n and for each r > 0

EKj;II(JJ.) =ETj;II(~) -t + 0 (J.L-') (JJ. -+ 00)

varKj;II(JJ.)=varTj;II(JJ.)+ 112

+ o (JJ.-') (JJ.-+oo).

Proof: Let j. n e IV with j S n. Then

- 12-

(14) P (Tj ;lIijJ.) > t) - P (TijJ.) > t) =Q (I ijJ., t» ,

where I ijJ., t) := P (T ijJ.) > t) and the polynomial Q is defined by

Q(X)=;~[;] (l-xl'x·-1-x.

Integration of (14) gives

00

(15) EI'j;lIijJ.) - EI'ijJ.) = f Q(I ijJ., t» dto

In a similar way we obtain

00

(16) E [Tj ;lIijJ.)] - E [TijJ.)] = :I: Q(I ijJ., k» .k=l

Now, since Q(O) =Q (1) =0, the polynomial Q(x) contains a factor x(1- x), and it follows from

an Euler-McLaurin-type result in [2] that the difference of the right-hand side of (15) and (16) is

OijJ.-r') ijJ.~ 00) for every r > O. By Corollary 1 the result on EKj;n(p.) can now be obtained

For the result on var Kj;n a similar proof applies. 0

Now Theorem 3 leads to the result we started out to obtain:

Theorem 4. For each j, n E IN with j $ n and for Il~ 00

1 t 2var Kj;nijJ.) = Il var Xj;n +"31l cov(Xj;n' Xj;n) + 0 (1) .

Theorem 4 has been proved in [1] by more laborious, purely analytic methods. The present proof

is a bit simpler, and fonnulas like (12) provide some more insight. Tables and asymptotic fonnu­

las for moments of Xj ;lI needed to apply Theorem 4 can be found in Harter [5]. It turns out that

the estimates are quite accurate even for moderate values of Il and fairly large values of n. The

first tenns (without error tenn) are easily obtained from the centrallimit theorem; the accuracy is

considerably increased by the extra tenns, but at a cost. Though the order tenns are not unifonn

in n, they can be shown to be quite good as long as n does not increase faster then polynomially

inll·

Acknowledgement: The authors are indebted to J.J.A.M. Brands for assistance with asymptotics

(especially with Lemmas 5 and 7) and to W.R. van Zwet for suggesting the method expressed by

equation (9).

- 13-

REFERENCES

1. Brands, J.J.A.M., Overdijk, D.A. and Steutel, F.W. (1986), Poisson order statistics.

Unpublished report Eindhoven University of Technology.

2. Brands, J.J.A.M. (1988), Asymptotics in Poisson order statistics, EUT-report 88-08, Dept.

of Mathematics and Computing Science, Eindhoven University of Technology, Eindhoven,

The Netherlands.

3. Breimann, L. (1968), Probability. Addison-Wesley.

4. Feller, W. (1971), An introduction to probability theory and its application, vol. 2, 2-nd.

ed., Wiley.

S. Harter, H.L. (1961), Expected values of nonnal order Statistics, Ann. Math. Statist. 48,

151-165.

6. Kolassa, J.E. and McCullagh, P. (1988), Edgeworth series for lattice distributions (pre­print).

7. Polak, P.W. (1987), Solution to problem 192. Statistiea Neerl. 41 (1),71-72.

8. Riordan, J. (1949), Inversion fonnulas in nonna! variable mapping. Annals Math. Statist.,

20,417425.

9. Steutel, F.W. and Thiemann, J.G.F. (1989), On the independence of integer and frac­

tional pans. Statistiea Neerl. 43 (1), 53-59.

10. Widder, D.V. (1972), The Laplace Transfonn, Princeton University Press.

,EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

PROBABILITY THEORY, STATISTICS, OPERATIONS RESEARCH AND SYSTEMS

THEORY

P.O. Box 513

5600 MB Eindhoven - The Netherlands

Secretariate: Dommelbuilding 0.03

Telephone: 040 - 47 3130

List of COSOR-memoranda - 1989

Number Month Author Title

M 89-01 January D.A. Overdijk Conjugate profiles on mating gear teeth

M 89-02 January A.H.W. Geerts A priori results in linear quadratic optimal control theory

M 89-03 February A.A. Stoorvogel The quadratic matrix inequality in singular H 00 control with state

H.L. Trentelman feedback

M 89-04 February E. Willekens Estimation of convolution tail behaviour

N. Veraverbeke

M 89-05 March H.L. Trentelman The totally singular linear quadratic problem with indefinite cost

M 89-06 April B.G. Hansen Self-decomposable distributions and branching processes

M 89-07 April B.G. Hansen Note on Urbanik's class Ln

M 89-08 April B.G. Hansen Reversed self-decomposability

M 89-09 April A.A. Stoorvogel The singular zero-sum differential game with stability using H 00 con-

trol theory

M 89-10 April L.J.G. Langenhoff An analytical theory of multi-echelon production/distribution systems

W.H.M.Zijm

M 89-11 April A.H.W. Geerts The Algebraic Riccati Equation and Singular Optimal Control

- 2-

Number Month Author Title

M 89-12 May D.A Overdijk De geometrie van de kroonwie1overbrenging

M 89-13 May !.J.B.F. Adan Analysis of the shortest queue problem

J. Wessels

W.H.M.Zijm

M 89-14 June A.A Stoorvogel The singular H 00 control problem with dynamic measurement feed-

back

M 89-15 June AH.W. Geerts The output-stabilizable subspace and linear optimal control

M.LJ. Hautus

M 89-16 June p.e. Schuur On the asymptotic convergence of the simulated annealing algorithm

in the presence of a parameter dependent penalization

M 89-17 July A.H.W. Geerts A priori results in linear-quadratic optimal control theory (extended

version)

M 89-18 July D.A Overdijk The curvature of conjugate profiles in points of contact

M 89-19 August A Dekkers An approximation for the response time of an open CP-disk system

1. van der Wal

M 89-20 August W.FJ. Verhaegh On randomness of random number generators

M 89-21 August P. Zwietering Synchronously Parallel: Boltzmann Machines: a Mathematical Model

E. Aarts

M 89-22 August !.J.B.F. Adan An asymmetric shortest queue problem

J. Wessels

W.H.M.Zijm

M 89-23 August D.A. Overdijk Skew-symmetric matrices in classical mechanics

M 89-24 September F.W. Steutel The gamma process and the Poisson distributionJ.G.F. Thiemann