106935 - zexazqda mixgap mi`yep

miwixt mii`xw` mipan

.iwqaepxb jexa :dvxn

:zextiqArratia, R., Barbour, A. and Tavare, S. (2004).

Logarithmic combinatorial structures: a probabilistic approach.Kolchin, V. (1999). Random graphs. Encyclopedia of mathematics and its applica-

tions. Cambridge Univ. Press.

Freiman, G., Granovsky, B.(2004) Clustering in coagulation-fragmentation process-es, random combinatorial structures and additive number systems:Asymtotic formulae

and limiting laws, Tr.AMS,v.357,6,

Granovsky, B., Stark, D. (2006) Asymptotic enumeration and logical limit laws forexpansive multisets and selections, J. London Math.Soc.,(2),73.

irah xtqn ly zewelg :`eanxehwe mi`xew n ∈ N+ := 0, 1, 2, . . . , xtqn ly dwelgl .1 dxcbd

η = (k1, . . . , kn) :n∑

j=1

jkj = n, kj ∈ N+.

( zeveaw =) mixaegn xtqn mi`xew |η| := k1 + . . . + kn-l.n ly zewelg agxn mi`xew Ωn = η-le η -a:o`k 10 = 3 + 3 + 1 + 3 = 2 + 4 + 4 :`nbec

η1 = (1, 0, 3, 0, . . . , 0) = 1133 ∈ Ω10

η2 = (0, 1, 0, 2, 0, . . . , 0) = 2142 ∈ Ω10.

.Y oung zenxb`ic i"r ozip zewelgd ly ixhne`b xe`zxehwe i"r zbvein ζ ∈ Ωn o`k

ζ = (l1, . . . , lm) : l1 ≥ . . . ≥ lm > 0 : l1 + . . . lm = n.

:miiwzne.dpey`xd dcenrd jxe` -|ζ| = m

1

.n ly zewelg ly llek xtqn |Ωn| := pn:onqp(.Euleur ly zewelg zxvei 't) :dprh

g(x) :=∑n≥0

pnxn =∏j≥1

(1− xj)−1, |x| < 1, p0 = 1.

:dgkedg(x) =

∏j≥1

∑

k≥0

xjk :=∑n≥0

cnxn. (1)

:oniq[xn]g(x) = cn

,(1) itl

[xn]g(x) = [xn]n∏

j=1

∑

k≥0

xjk =

[xn]( ∑

k1≥0,...,kn≥0

x1k1+2k2+...+nkn

)=

∑

(k1,...,kn)∈Ωn

1 =

pn.

miwixt mipanmlceby miwixt-i` mipan n ly cegi` `ed n lcebn dpan leky jk mipana opeazp

xnelk ,k1, . . . , kn

n =n∑

j=1

jkj.

wixt dpan ly miiqiqa mipiit`n;n lcebn wixt dpan -πn

;n lcebn miwixt mipan xtqn- pn

;j lcebn wixt-i` dpan ly (mibeq=) miavn xtqn-mj

.πn oezp dpan ly (zeipn mexhwtq=)mexhwtq -η = (k1, . . . , kn) ∈ Ωn

cg cg `l llk jxca `idy η ∈ Ωn mexhwtq ← πn dpan dn`zdd zniiw :dxrd.zikxr

:ze`nbec.[n] = 1, 2, . . . , n lr zexenz .`

Sn = πn, |Sn| = n!

.πn = (πn(1), . . . , πn(n)) (mipan =)zexenzd lk ly sqe` -`a 1 ixg` ,1-n ligzn oey`x lbrn :onwlck milbrn ly cegi`k meyxl ozip dxenz lkycg lbrn miligzn ,miizqn oey`xd lbrndy ixg` .'eke πn(πn(1)) eixg` ,πn(1) xtqn

,lynl .d`ld jke,exzepy mixtqnd oia xzeia ohwd xtqndn

π8 = (84357612) = (182457)(3)(6)

2

.π8-d ly zilbrn dbvd-,miwixt -i` (miaikx =)mipan md milbrn o`k.j lcebn milbrnd xtqn -mj = (j − 1)! -y jk

eze` zelra πn zexenz N(k1, . . . , kn) xtqnzgqep i"r ozip η = (k1, . . . , kn) ∈ Ωn mexhwtq

:(Cauchy)iyew

N(k1, . . . , kn) = n!n∏

j=1

(1

j

)kj 1

(kj)!.

.mj = (j − 1)!-e mepihlen zgqep :(fnx)dgked:libxz

π8 = (84357612) = (182457)(3)(6) →

η = (2, 0, 0, 0, 0, 1, 0, 0).

,iyew zgqep itl .N(2, 0, 0, 0, 0, 1, 0) `vnp

N(2, 0, 0, 0, 0, 1, 0, 0) = 8!(12 1

2!

1

6· 1

)

= 3 · 4 · 5 · 7 · 8 = 3360.

: Sn-a zi`xw` dxenz ly dincd(oey`x lbrnl sxhvn=) 1 ly epinin mewn qtez 2 xtqn.oey`x lbrn ligzn 1 xtqnligzn k xtqny jk jynp jildzd .ycg lbrn ligzn 1/2 zexazqdae 1/2 zexazqdacg` lk ly epinin mewn qtez `ed 1/k zexazqd dze`a e` 1/k zexazqda ycg lbrn

miiwzn .dxvepy zi`xw`d dxenzd πn onqp .el encw xy` mixtqn (k − 1) -n

P (πn) = 1(1/2)(1/3) . . . (1/n) =1

n!.

zellweyn zewelg .a, j ieeyd eze`a zerahn ly mibeq mj mpyiy migipn

i"r n llek mekq ly zewelga opeazp .j = 1, 2, . . . xy`kzewelgd xtqn- pn idi.cg`k mbeqae zerahn ly mieeya dielz dwelg lk o`k.l"pd zerahnd

`id pn ly zxvei 't f` .n ly zellweyn

g(x) =∏j≥1

(1− xj)−mj , |x| < 1

mitxb.b(miwixt-i` mipan)mixiyw mitxb xtqnl wxtzn (n lceba dpan=)micewcew n lr sxb lk

:miiwzne micewcew j lr mixiywd mitxbd xtqn `ed kj-y jk

η = (k1, . . . , kn) ∈ Ωn.

3

o`kpn = 2(n

2).

GF (qn) iteq dcy lrn n dwfgn minepilet.cidi .ipey`x xtqn `ed q dcyd (oiivn=)oiite`y xikfp

f(x) = xn + an−1xn−1 + . . . + a0

.pn = qn if` .GF (qn) iteq dcy lrn mepilet`ed kj y jk , GF (qn)-a miwixt-i` minepilet ly dltknk bivdl ozip l"pd f mepilet lk

.η = (k1, . . . , kn) ∈ Ωn -e j dwfga miwixt-i`d minxebd xtqnn = 6, q = 2 xear ,lynl

f(x) = x6 − x5 + x4 − x2 + x− 1 = (x− 1)3(x + 1)3.

wexite befn ly miihqkehq mikildz.d.η =-e j lceba zeveawd xtqn kj -y jk (miyiab=)zeveawl miwlegn midf miwiwlg n

(k1, . . . , kn) ∈ Ωn

`ly e` oezp rbx lkay jk ziccd zelret zeveawd .η i"r xcben oezp rbx lka zkxrnd avn:mi`ad mirexi` ipyn cg` ygxzn e` xac ygxzn

zg` dveawl zebfnzn j-e i mlceby zeveaw izy :befn •xnelk ,i + j dlceby

η = (k1, . . . , kn) → η(i,j) =

(k1, . . . , ki − 1, . . . , kj − 1, . . . , ki+j + 1, . . . , kn).

zeveaw izyl zwxtzn i + j dlceby zg` dveaw :wexit •xnelk ,j-e i mlceby

η = (k1, . . . , kn) → η(i,j) =

(k1, . . . , ki + 1, . . . , kj + 1, . . . , ki+j − 1, . . . , kn).

geqip :mipand zxitq ly zeihehtniq` zeira;n →∞ -yk pn ly zihehtniq` zebdpzd .`

,dpezp mipan dveaw zz An ⊂ πn idz .a.znieqn dpekz miniiwnd πn mipan ly sqe` `id An xnelk

.n →∞ -yk , |An| ly zihehtniq` zebdpzd `vnl yi:πn-a An ly zetitv xicbp .zizexazqd dhiy `id oexztl zelirid zehiydn zg`

0 ≤ ρn :=|An|pn

≤ 1.

4

if` ,mipand lk ly πn sqe`d lr dcig` dcin U m`

ρn = P (An).

ze`nbece dxcbd :mi`xw` miwixt mipanxnelk , πn mipand sqe` lr dcig`d zebltzdd U idz :2 dxcbd

U(πn) =1

|πn| =1

pn

, πn ∈ πn.

.`.ele i`xw` dpan mi`xew xvep xy` Πn i`xw`d hpnl`l if`

K(n) = K(n)(Πn) = (K(n)1 (Πn), . . . , K(n)

n (Πn)),

j lceba miwixt-i` miaikx xtqn- K(n)j (Πn) yk

.Sn-a zexenz :`nbec .Πn i`xw` dpan ly mexhwtq mi`xew , Πn-a,iyew zgqep itl

P(K(n) = (k1, . . . , kn)

)=

N(k1, . . . , kn)

pn

=

n∏j=1

(1

j

)kj 1

(kj)!, (k1, . . . , kn) ∈ Ωn.

.πn-a mikxr lawn Πn i`xw` dpan df znerl .Ωn-a mikxr lawn K(n) .`.e :dxrddpzen xyw

-a mikxr ilrae .z.a .`.n ly dxciq- Zj, j ≥ 1 idz

N+ := 0, 1, 2, . . . , ,-y jk

P (Zj = k) = a(j)k , k ∈ N+, j ≥ 1.

.`.e ly dxciq xicbpK(n) = (K

(n)1 , . . . , K(n)

n ), n ≥ 1:`ad ote`a Zj, j ≥ 1 dxciqd i"r zxvepd

L(K(n)) = L(Z1, . . . , Zn|n∑

j=1

jZj = n), n = 1, 2, . . . ,

:miiwzn

P(K(n) = (k1, . . . , kn)

)=

∏nj=1 a

(j)kj

P (∑n

j=1 jZj = n),

η = (k1, . . . , kn) ∈ Ωn, n ≥ 1.

5

onqp

cn := P (n∑

j=1

jZj = n) =∑η∈Ωn

n∏j=1

a(j)kj

,

η = (k1, . . . , kn)

: Ωn iteq agxn lr µn zizexazqd dcin dzr xicbp

µn(η) := c−1n

n∏j=1

a(j)kj

, η = (k1, . . . , kn) ∈ Ωn. (2)

.dcind ly dwelg zivwpet z`xwp cn o`k-"zeieihwiltihlen" mi`xew (2) dxevdn zecinl

(V ershik, 1996).(Pitman, 2002) -Gibbs zecin e`

.K(n) ⇔ µn miiwznmiwixt mipan ly ce`n agx oeebn ly mexhwtqy xazqn

: Zj .`.n ly ze`ad zeiebltzdd yely k"dq i"r lawzn mii`xw`.inepiae ilily inepia ,ipeq`et

mexhwtq ly zeillk zepekz :1 dprh=) zertzydd zivnxetqpxh zgz ihp`ixepi` `ed (µn dcind=) K(n) mexhwtqd .`

: Zj, j ≥ 1 .`.n ly (tilting

a(j)k (θ) = bjθ

jka(j)k , k ≥ 0, j ≥ 1, θ > 0

.lenxip reaw bj = bj(θ)− -ykidi .Kolchin ly dbvd .a

Ωn,k := η ∈ Ωn : |η| = k, 1 ≤ k ≤ n.

miiwzn η ∈ Ωn,k lk xear if`

P(K(n)(Πn) = η||η| = k

)= c−1

n,k

n∏j=1

a(j)kj

,

1 ≤ k ≤ n, (3),

dwelg zivwpet mr Ωn,k lr qaib zcin `id l"pd zipzend zexazqdd xnelk

cn,k =∑

η∈Ωn,k

n∏j=1

a(j)kj

.

6

.zicin .` :dgked-l deey oecipay zipzend zexazqdd .a

P(K(n)(Πn) = η

)

P(|K(n)(Πn)| = k

) ,

η ∈ Ωn,k. (4)

o`kP

(|K(n)(Πn)| = k

)=

∑η∈Ωn,k

P(K(n)(Πn) = η

):=

cn,k (5)

dpan ly iyteg xhnxt mi`xew θ-l .` :zexrd .(5)-e(2) z` (4)-a aivdl x`yp zrk.okl mcew exkfed xy` zeihehtniq` zeira xwga aeyg ciwtz wgyn `ed .i`xw`

-iltihlen dcinn zxfbpd Ωn,k lr zipzend zexazqdd ,oezp k lk xeary raep (3)-n .a.ziaihwiltihlen dcin ok enk `id µn ziaihw

mi`xw` mipan ly miixwir mibeq dyelyexkfedy Zj, j ≥ 1 .`.n ly zeiebltzdd yelyl m`zda

:mi`xw` miwixt mipan ly mi`ad mibeqd zyelya milican ,okl mcewe(Multisets =) zeveaw ax(Assemblies=)mitq`

.mixgan(Selections=):d`ad zkxrn dze` "ir ilniq ote`a x`zl ozip mipand ly mibeqd zyely lk z`

n lceba wixt dpan

(mn) wixt-i` . . . (m2) wixt-i` (m1) wixt-i`,n lceba miwixt-i` mipan kn-n,. . . ,1 lceba miwixt-i` mipan k1 -n akxen πn dpan lk,mixtqenn md miwiwlgd zetiq` ly dxwna ik d`xp jynda .(k1, . . . , kn) ∈ Ωn -y jkwiwlgn xzei `l qpkp `z lka Selections ly dxwnae midf md Multisets ly dxwna

.cg`ly zexvei zeivwpete minexhwtqd zeiebltzd z` lawp ep`

.mii`xw` mipan ly mibeqd zyely xear pn zexcqeidi .zewfg ixeh xear iyew zltkn xicbp ziy`x

f1(x) =∑i≥0

tixi, f2(x) =

∑j≥0

rjxj,

. mdly zeqpkzdd qeicx oeiv `ll xnelk ,"miilnxet" zewfg ixeh ipy:onwlck zxcben mdly f = f1 ∗ f2 iyew zltkn if`

[xn]f(x) =n∑

i=0

tirn−i, n ≥ 0.

7

cn ly zxvei divwpet :2 dprhly zeiexazqd zxvei zivwpet S(j)(x) =

∑k≥0 a

(j)k xkj idz

zivwpet `id g =∏

j≥1 S(j) iyew zltkn if` .Zj .`.n.cn dxciqd ly zxvei

:dgked

[xn]g(x) = [xn]∞∏

j=1

S(j)(x) = [xn]n∏

j=1

S(j)(x) =

∑η∈Ωn

n∏j=1

a(j)kj

= cn.

- y dgpda .mcew ebvedy milceb dnk ly dcin dpw zepyl jxhvp ep`

a(j)0 = P (Zj = 0) > 0, j ≥ 1,

xicbp

a(j)k :=

a(j)k

a(j)0

, j ≥ 1

xicbp z`fl m`zdae

cn =( n∏

j=1

a(j)0

)−1cn, n ≥ 1, c0 = 1.

dn`zda od cn-e a(j)k zexcq ly zexveid zeivwpetd okl

:g -e S(j)

S(j)(x) =S(j)(x)

a(j)0

, j ≥ 1,

g(x) =∏j≥1

S(j)(x).

• (scaled=)minleqn milceb zxfra meyxl ozip mexhwtqd zgqep z`y ze`xl lw dzr:onwlck

P(K(n) = (k1, . . . , kn)

)=

∏nj=1 a

(j)kj

cn

,

η = (k1, . . . , kn) ∈ Ωn, n ≥ 1. (6)

zyely lkay d`xp jynda .znleqn dwelg zivwpet `xwl ozip cn-ly raep l"pd meyixn.pn ly zxvei divwpet `id g ,mi`xw` mipan ly l"pd mixwnd

.zetiq`: aj xhnxt mr ipeq`et bletn Zj dfd dxwna

okl .Zj ∼ Po(aj), j ≥ 1

a(j)k = e−aj

akj

k!, j ≥ 1, k ≥ 0.

8

,okl

a(j)k =

akj

k!, j ≥ 1, k ≥ 0

-e

P(K(n) = η

)= (cn)−1

n∏j=1

akj

j

kj!, η ∈ Ωn, (8)

xy`k

cn =∑η∈Ωn

n∏j=1

akj

j

kj!. (9)

miwiwlg mr zkxrnl dni`zn (8) zebltzdd ik d`xp dzrη = (k1, . . . , kn) ∈ Ωn idz .mixtqenn

,miwixt-i` mipan oia mixtqenn miwiwlg n ly dpezp dwelgz`y xne` df .miwiwlg j "miqpkp" j lcebn miwixt-i`d mipand kj-n cg` lkly jk

mipte` xtqna rval ozp η l"pd dwelgd

n!

(1!)k1k1!(2!)k2k2! . . . (n!)knkn!

lkl ,j lceba wixt-i` mipan ly mibeq mj mpyiy meyn.zinepihlen dgqep "ir lawzndzeni`zn l"pd mibelitdn cg`

mk11 . . . mkn

n

kd"q eidi xac ly enekiqa .zepey ze`qxib

n!n∏

j=1

(mj

j!

)kj 1

kj!

:`ed n lcebn mipand ly llekd mxtqn ,jkn d`vezk.l"pd η mexhwtq ilra πn mipan

pn = n!∑η∈Ωn

n∏j=1

(mj

j!

)kj 1

kj!.

onql x`yp dzraj =

mj

j!, j ≥ 1

-y lawl ickpn = n!cn (10)

.(8) ly ipni sb`l deey ρn = P (An) ,η mexhwtq ilra πn mipand zveaw- An xearye.(10) zxfra pn-l dxeywd cn ly zxvei divwpet dzr `vnp

S(j)(x) =∑

k≥0

e−ajak

j

k!xjk = e−aj

∑

k≥0

(ajx

j)k

k!=

9

e−aj exp (ajxj), j ≥ 1

-y jk a(j)0 = e−aj o`k

S(j)(x) = exp (ajxj), j ≥ 1,

-eg(x) = exp (

∑j≥1

ajxj). (11)

.zetiq` ly ze`nbec-e aj = 1

j-y jk ,mj = (j − 1)!-y epi`x .zexenz .`

g(x) = exp (∑j≥1

1

jxj) =

exp(

log1

1− x

)=

1

1− x=

∑n≥0

xn.

.iyew zgqepl m`zda ,cn = 1, n ≥ 0 okl.mexhwtqd ly zeileyd zeiexazqdd -P (K

(n)j = kj) z` aygp

z` xicbp .zay zcewp s` dlikn `l zi`xw` dxenzy zexazqdd-P (K(n)1 = 0)-n ligzp

rxe`nd

E(n)i = πn ∈ Sn : πn(i) = i.

if`

P (K(n)1 = 0) = 1− P

( n⋃i=1

E(n)i

). (12)

dgcd-dlkd oexwr itl

P( n⋃

i=1

E(n)i

)=

n∑i=1

P(E

(n)i

)−

∑1≤i1<i2≤n

P(E

(n)i1

⋂E

(n)i2

)+ . . .

(−1)l+1∑

1≤i1<...il≤n

P(E

(n)i1

. . .⋂

E(n)il

)+ . . .

(−1)n+1P(E

(n)1

⋂. . .

⋂E(n)

n

).

raep dfn

P (K(n)1 = 0) = 1−

(n

(n− 1)!

n!− n(n− 1)

2

(n− 2)!

n!+ . . .

10

(−1)l+1

(n

l

)(n− l)!

n!+ . . . (−1)n+1 1

n!

)=

n∑

l=2

(−1)l

l!=

n∑

l=0

(−1)l

l!→ e−1, n →∞.

,dnec ote`a

P (K(n)j = k) =

j−k

k!

[nj]−k∑

l=0

(−1)l j−l

l!→

exp (−1

j)

(1j)k

k!= Po(

1

j), n →∞. (13)

.Zj, j ≥ 1 .`.n ly zebltzdl ddf zileabd zebltzddy yibcply zillk dpekz ly ihxt dxwn `id (13)-y d`xp cizra

zeiaihwiltihlen zecin ly zihehtniq` zebdpzd.(minexhwtq=)

Ewens(1972) ly mbcnd zgqep .a.(Ewens sampling formula = ESF )

:lcend ly zilnxet dxcbdaj =

θ

j, θ > 0, j ≥ 1.

if`

cn =[exp(

∞∑j=1

θ

jxj)

]n

= [exp (−θ log(1− x)]n =

[(1− x)−θ]n = (−1)n (−θ)(−θ − 1) . . . (−θ − n + 1)

n!=

θ(θ + 1) . . . (θ + n− 1)

n!:=

θ(n)

n!

: Ewens zebltzdl miribn ep`e

µn(η) =

( n!

θ(n)

) n∏j=1

(θ

j

)kj( 1

kj!

)=

( n!

θ(n)

)θ|η|

n∏j=1

1(jkj

)kj!

,

η = (k1, . . . , kn) ∈ Ωn. (14)

11

ly `ad lcenk drited Ewens zebltzd xewnaob lkl .seb i`z oia mixfetnd mipb ly n ax xtqn epyi mc` oa lkl :oiqelke` zwihpb:zeivhen ly xtqn eze` ilra mipbd zeveawl wlgzn n-y jk ,(zeivhen=)zexev dnk okziθ xhnxtd l"pd lcena.cg` lk zeivqen j ilra mipbd mxtqn df kj xy`k n =

∑nj=1 jkj

.Ewens zebltzd zincdl jxc dlbn d`ad dprqd .divhend avw `han-y jk zi`xw` dxenz Π

(θ)n ∈ Sn idz dprq

P (Π(θ)n = πn) =

θ|πn|

θ(n), θ > 0, πn ∈ Sn, (15)

.Ewens zgqep itl bletn Π(θ)n ly mexhwtqd if` .πn-a milbrn xtqn -|πn| xy`k

zelra πn ∈ Sn zexenzd lk zveaw An(η)-ae Π(θ)n ly mexhwtq K(n,θ) onqp :dgked

okl . |η| jxr eze` zelra od πn ∈ An(η) lk ,(15) itly oiivp .η ∈ Ωn mexhwthd eze`miiwzn

P(K(n,θ) = η

)=

∑

πn∈A(η)

P (Π(θ)n = πn) = N(η)

θ|πn|

θ(n),

.l.y.n ,iyew htyn itl,okl .N(η) = |An(η)| xy`kzi`xw` dtiq`e Π

(θ)n zi`xw` dxenz :mii`xw` mipan ipy ly minexhwtq dpwqn

.ddf ote`a mibletn mj = θ(j − 1)! mrEwens zebltzd ly lcenk zipiq dcrqn

,zepgleydn cg` ci-lr ayizn oey`xd gxe`d .milebr zepgley seqpi` zipiq dcrqnazexazqda oey`xd ly epinin ayiizn e` θ

θ+1zexazqdd mr ycg ogley qtez ipyd gxe`d

ayiizn 1θ+k−1

zexazqdae θθ+k−1

zexazqda ycg ogley qtez k ≥ 2 xtqn gxe`d . . . , 1θ+1

dbvd zxvep n xtqn gxe` ly e`ea ixg` .mincewd migxe`d k − 1-n cg` lk ly epininmiaygp ogley eze` ci-lr miayeid migxe`d xy`k n ly zi`xw` dxenz ly zilbrn.oey`x eze` qtez xy` gxe`d xtqn itl mixceqn(milbrn=)zepgleye lbrn eze`l mikiiy

if`

P (πn) =θ|πn|−1

(θ + 1) . . . (θ + n− 1)=

θ|πn|

θ(n).

.Π(θ)n zi`xw`d dxenzd zxvepy xne` df

:`ad oet`a eayiizi migxe` 6-y zexazqdd idn :libxz

π6 = (1, 6)(2, 4, 5)(3).

-l deey zyweand zexazqdd :oexzt

P (π6) =( 1

θ + 5

)( θ

θ + 1

)( 1

θ + 3

)( 1

θ + 4

)( θ

θ + 2

)=

θ3

θ(6)

:zexrd-end z` z`han ycg ogley ci-lr k xtqn gxe`d zeayiizd zipiqd dcrqnd lcena .`

.k-d xeca obd ly divh

12

:zihpbd dyxezd iweg ze`xwpd ze`ad zegpdd izy ly qiqa lr dlawzd ESF .a;mincewd zexecd (k−1)-a mipbd zebltzda dielz `l k-d xeca divhend zexazqidd (i),k-l encw xy` zexeca minrt kj ritedy j beql jiiy k-d xeca obd beqy zexazqdd (ii)

.cala kj-e j-a dielzn-a avnd zpenz idze n = 9 idi .dnbecd i"r l"pd dyxezd iweg meiw xiaqp libxz

:onwlck (zipiq dcrqna zepgleyd oia migxe`d n-d xefit =)mincewd zexec2, 3, 8 1

5, 7, 9 4

6η = (3, 0, 2, 0, . . . , 0) ∈ Ω9 mexhwtq mi`zn z`fk daiyil

(1 beqn mipb=)zg` divhen ilra mipb xnelk ,etvpy zexecd 9-a mipbd zwelg xicbn xy`ilra mipbe minrt 3 etvp

?3 beqn idi 10 xeca oby zexazqdd idn .miinrt etvp (3 beqn mipb=) zeivhen yelyipyn cg` ci -lr ayiizi 10 xtqn gxe`dy zexazqdl deey zyweand zexazqdd:daeyz

-l deey `id okl .cg` lk migxe` 3 mr zepgley

3k3

θ + n=

6

θ + 9,

.(ii) wegl m`zga.zeigd zebdpzd ly megzn ESF -d yext ok enk `iap

.dign ghy lr zexgzghy lr mixgzn zeig ly E1, E2 . . . , zepey(zeveaw=)mipin

-dn cg`l zkiiyd zg` dig "ir mrt lk yakp ghyd 1, 2, . . . , mipnfay jk ,miieqnxehwed i"r zrawp ghyd z` eyak xy` zeigd n oia mipind zwelg .l"pd mipinj ghyd z` eyak xy` E1, E2 . . . , oia mipind xtqn `ed kj xy`k η = (k1, . . . , kn) ∈ Ωn

.ghya mibivp(n jezn) j mdl yi xnelk minrt,ycg oinn dig i"r yakp ghyd n onfay zexazqdd m` dprh

-l deeyθ

θ + n− 1, θ > 0

.milewy dyexi weg ly (ii)-e (i) mi`pzd if` ,jk ,Ek oinn zeig nk ,. . . , E1 oinn zeig n1 etvp ghya n onf lleke cry gipp :dgkeddig i"r yaki ghyd n + 1 onfay zexazqdd z` p(ni, n)-a onqp .n1 + . . . + nk = n -y

-y jkn raep .Ei, 1 ≤ i ≤ k oinnk∑

i=1

p(ni, n) +θ

θ + n= 1. (16)

,(16)-n lawp n1 = . . . = nn = 1-e k = n xear

p(1, n) =1

θ + n.

mewna n1 − 1 ,(16)-a gwipe n1 > 1 -y lynl gipp ,ni > 1-n cg` zegtl (16)-ay meyn:iepiy `ll ni-d xzie nk+1 = 1 ,n1

13

p(n1 − 1, n) + p(n2, n) + . . . + p(nk, n)+

p(1, n) +θ

θ + n= 1. (17)

:(16)-n (17) zigtp

p(n1, n)− p(n1 − 1, n) = p(1, n) = p(n1, n) =1

θ + n.

oklp(n1, n) =

n1

θ + n,

.n, n1 ≤ n lk xear.l.y.n .ziciin dketdd dprhd ly dgked

:dpwqn.ESF zebltzdd itl bletn ,mipind oia zi`xw` dwelgd z` xicbnd K(n,θ) .`.e

K(n,θ) -a zeveawd xtqn|K(n)| = ∑n

j=1 K(n)j .`.nd z` xicbp edylk Πn wixt i`xw` dpan ly K(n,θ) mexhwtq xear

milbrn xtqn df zexenzd dxwna .Πn-a (zeveaw=) miwixt-i` mipand llek xtqn `edyghya etvpy mipeyd mipind xtqn e` mipeyd mipbd ibeq xtqn df ESF ly dxwnae

.n onf lleke cr daixnmilbrn k zelra n-n zexenzd xtmn :dnl

idiSn,k := πn ∈ Sn : |πn| = k, cn,k := |Sn,k|, k ≤ n.

:miiwzn(i.) c(n, k) = (n− 1)c(n− 1, k) + c(n− 1, k − 1),

1 ≤ k ≤ n, c(0, 0) = 1, c(n, k) = 0, n, k ≤ 0,

(ii.)n∑

k=0

c(n, k)xk = x(x + 1) . . . (x + n− 1) = x(n).

zexenz (n − 1) zeni`zn πn−1 ∈ Sn−1,k zexenzdn zg` lkl .n ≥ k ≥ 1 idi .(i)dgkedπn−1-dn zelawznd π′n ∈ Sn,k

mixtqndn cg` ly epinil n xtqnd ztqed i"r.π′n(n) 6= n-y xirp .1, 2, . . . , n− 1

(n− 1)cn−1,k zelawzn jkn d`vezkdxenz lkl jci`n .Sn,k-l zekiiyd l"pd beqn zexenz

zg` dxenz dni`zn πn−1 ∈ Sn−1,k−1

π′n ∈ Sn,k : π′n(n) = n,

14

:`ad ote`a zlawznd

π′n(i) =

πn(i), `m i 6= n

n, `m i = n.

.(i) epgked ,Sn,k-a zexenzd lk z` epxtq xac ly enekiqay llba.(i) i`pzd z` miniiwn b(n, k) mincwndy `cep .x(n) =

∑nk=0 b(n, k)xk idi.(ii)

miiwzn

x(n) = (x + n− 1)x(n−1) =n∑

k=1

b(n− 1, k − 1)xk+

(n− 1)n−1∑

k=0

b(n− 1, k)xk

⇒ (ii)

.l.y.n.oey`x beqn Stirling ixtmn mi`xew s(n, k) := (−1)n−kc(n, k), k ≤ n mixtqnl :dxrd

:miiwznn∑

k=0

sn,kxk = x(x− 1) . . . (x− n + 1) := x(n).

:miiwzn :dprh

(i.) P(|K(n,θ)| = k

)=

θk

θ(n)cn,k

-e

(ii.)E(|K(n,θ)|

)=

n∑

k=1

θ

θ + k − 1.

:dgked(i)P

(|K(n,θ)| = k

)=

∑

πn:|πn|=k

P (Π(θ)n = πn) =

θk

θ(n)c(n, k).

idze Z := |K(n,θ)| onqp (ii)

gZ(x) =n∑

k=0

P (Z = k)xk = ExZ

miiwzn ,(i) jnq lr .Z .`.n ly zeiexazqd zxvei divwpet

ExZ =n∑

k=0

xk θk

θ(n)c(n, k) =

1

θ(n)

n∑

k=0

(xθ)kc(n, k) =

15

(xθ)(n)

θ(n)=

n∏

k=1

(1− θ

θ + k − 1+

xθ

θ + k − 1

)=

n∏

k=1

(xpk + qk

), (18)

zxvei divwpet mr ddcfn xpk + qk divwpety meyn .qk = 1 − pk -e pk = θθ+k−1

xy`kBe(pk), k = .`.n n ly mekqk bletn Z .`.ny (18)-n raep ,Be(pk) .`.n ly zeiexazqd

.l.y.n .EZ =∑n

k=0θ

θ+k−1okl .miielz izla 1, . . . , n

xtqn dwihehtniq`d z` lawl xyt`n dprhd ly (i) wlgxeary (Moser,Wyman, 1958) reci.ESF -a zeveawd

:miiwzn k = o(log n)

c(n, k) ∼ (n− 1)!(γ + log n)k−1

(k − 1)!, n →∞,

,okl.dreci reaw γ xy`k

P(|K(n,θ)| = k

)∼ θk

θ(n)(n− 1)!

(γ + log n)k−1

(k − 1)!,

k = o(log n), n →∞.

.ilwel leab htyn ly dnbec ef(CFP ) mikitd wexite befn ikildz .bzekitd zecine zeihp`ixepi` zecin:`ean

befn jildz ,t ≥ 0 onf ly rbx lka ,(CFP ) zxcbd itllk jxe`l Ωn-a mikxr lawn CFP xnelk ,η ∈ Ωn dwelg i"r x`ezn (CFP =) wexite:mixarn ipy ly mdiavw xicbdl jixv ,onfa CFP ly mikxrd zepzyd reawl ick.t onfd

η → η(i,j), η → η(i,j)

el`d (mixarnd zeiexidn=)mixarnd iavw .i, j : 2 ≤ i + j ≤ n lke η ∈ Ωn lk xear,Ψ(i, j; η) xn`p ,oey`xd xarnd avw if` .t onfa jildzd eavn Xt-a onqp .onwlck mixcben

:jk xcben

Ψ(i, j; η) := lim∆t→0

P(Xt+∆t = η(i,j)|Xt = η

)

∆t.

:Φ(i, j; η) xn`p,ipyd xarnd avw z` mixicbn dnec ote`a

Φ(i, j; η) := lim∆t→0

P(Xt+∆t = η(i,j)|Xt = η

)

∆t.

t ≥ 0 lk xear zetwz l"pd zexcbdd izy.i, j : 2 ≤ i + j ≤ n lke

16

.jildzd ly dwinpicexwind z` zexicbn Φ-e Ψ zeivwpetd,ipexwr ote`a ,zexyt`n l"pd zeivwpetd izyy reci miihqkehq mikildz ly zillk dxeznlka Ωn agxn lr jildzd ly µt(η) zebltzd `idy jildzd ly dwinpicexwn mb zelbl

xnelk ,t ≥ 0 onfµt(η) := P (Xt = η), η ∈ Ωn, t ≥ 0.

ofe`n avnl ribn jildzd t = ∞ onfay reci ok enk: Ωn lr µ ,(zihp`ixepi` dcin = zpfe`n zebltzd =)

µ(η) := limt→∞

µt(η).

xy`k c(η, η) = 0-y xexa .η → η xarnd avw c(η, η), η 6= η ∈ Ωn-a onqp

η ∈( ⋃

i,j

η(i,j))( ⋃

i,j

η(i,j)

), η 6= η.

-y xexa ok enkP

(Xt+∆t = η|Xt = η

)→ 1, ∆t → 0.

:`ad ote`a ,η-n d`ivid avw-c(η, η) xicbp jk meyn

c(η, η) = lim∆t→0

1− P(Xt+∆t = η|Xt = η

)

∆t.

-y jk dl df mixeyw d`ivid avwe xarnd iavwy oiivp

c(η, η) =∑

η 6=η

c(η, η), η ∈ Ωn. (19)

-y dcaern raep df∑

η 6=η

P(Xt+∆t = η|Xt = η

)=

1− P(Xt+∆t = η|Xt = η

).

:zniiwn `idy jka zpit`zn µ zihp`ixepi`d zebltzdd∑

η 6=η

c(η, η)µ(η) = c(η, η)µ(η), η ∈ Ωn. (20)

"mxf"l deey (20) ly il`nyd sb`d :oefi` z`eeyn yext .oefi` z`eeyn z`xwp (20)- d(ofe`n=)icinz avnay xne` oefi`d i`pz .η ∈ Ωn : η 6= η lkn η-l qpkpd llekd izexazqd

.((20) ly ipnid sb`d=) η -n `veiy "izexazqd "mxf"l deey zeidl jixv dfd mxfdi`pzd z` miniiwn c(η, η) iavwe µ dciny dzr gipp

µ(η)c(η, η) = µ(η)c(η, η), η, η ∈ Ωn (21)

17

bef lk xeary (21)-d yext.hxetnd oefi`d i`pz `xwpdz`xwp (21) zniiwn xy` dcind.mieey md η → η-e η → η "minxf" ipy ,η, η ∈ Ωn lyjnq lr ,mpn` .(20) xxeb (21)-y xnelk ,zihp`ixepi` `id dkitd dcin lky d`xp .dkitd

:(21), (19)∑

η 6=η

c(η, η)µ(η) =

∑

η 6=η

c(η, η)µ(η) = µ(η)∑

η 6=η

c(η, η) = µ(η)c(η, η).

.l.y.n:d`ad dxevdn md CFP ly xarnd iavwy gipdl irah

Ψ(i, j; η) := Ψ(i, j; ki, kj) = kikjψ(i, j),

i 6= j, 2 ≤ i + j ≤ n,

Ψ(i, i; η) := Ψ(i, i; ki, ki) = ki(ki − 1)ψ(i, i),

2 ≤ 2i ≤ n,

Φ(i, j; η) := Φ(i, j; ki+j) = ki+jφ(i, j), 2 ≤ i + j ≤ n.

iavwk dn`zda miyxetn ψ(i, j), φ(i, j) o`kiavwk xnelk ,zeveaw izyl zccea dveaw ly wexit iavwe zeccea zeveaw izy ly befndxvep xy` idylk dtiq` ly mexhwtql dni`znd dcin µ idz :dprh ."miilebq" xarn

.Zj ∼ Po(aj), j ≥ 1 i"r:miniiwnd miilebq xarn iavwl qgia dkitd µ if`

q(i, j) :=ψ(i, j)

φ(i, j)=

ai+j

aiaj

, 2 ≤ i + j ≤ n.

:miiwzn (8) itl .hxetnd oefi`d i`pz meiw `ceep:dgked

µ(η) = (cn)−1

n∏j=1

akj

j

kj!, η ∈ Ωn.

:miiwzn i 6= j : i, j ≥ 1 xeary jkn raep

µ(η(i,j)) = µ(η)(kjki

ajai

)( ai+j

ki+j + 1

).

k meyxl ozip z`f(ki+j + 1)aiajµ(η(i,j)) = kjkiai+jµ(η).

18

xarn iaviwl qgia hxetnd oefi`d i`pz edf

c(η(i,j), η) = Φ(i, j; η(i,j)) = (ki+j + 1)aiajl(i, j)

-ec(η, η(i,j)) = Ψ(i, j; η) = kjkiai+jl(i, j)

.idylk divwpet l(i, j) > 0 xy`k-nxtd ,CFP -l xywda :dxrd .mixzepd mixwind ipya dprhd z` migiken dnec ote`a

dxevn mikxr `weec e`l xnelk ,mdylk miilily-i` mikxr lawl mileki aj mixh

aj =mj

j!, j ≥ 1

.ziteq dveaw ly zewelg.c

n ly ziteq dveaw [n] = a1, . . . , an-a onqp.[n] = 1, . . . , n-y gipdl xyt` dllkdd zlabd ila .[0] = Ø xy`k ,a1, . . . , an mixai`

Ci ⊆ [n], i = zeveaw zz ly dveaw `id [n] ly πn = (C1, . . . , Cl), 1 ≤ l ≤ n dwelg-y jk 1, . . . , l

Ci 6= Ø, j = 1, . . . , l •Cj

⋂Ci = Ø, j 6= j •

.C1

⋃C2

⋃. . .

⋃Cl = [n] •

.πn-a miwela xtqn |πn| = l-a mipnqne miwela mi`xew Ci-lπn = wixt dpan lkly jk ,Ci welad lceb df |Ci|-e miwixt-i` mipan md miwela o`k

(C1, . . . , Cl)πn-a miwelad xtqn `ed kj xy`k (mexhwtq=) , η = (k1, . . . , kn) ∈ Ωn dwelgd dni`zn

zeaiyg oi`y oiivp.j mlcebymiwelady (1, 2 . . . , n jezn) ilnipin xtqn itl mze` xtqnl mkqed .miwelad zniyx xcql

.milikn.eze` miaikxnd mixai`a wx oiit`zn wela lk ,dirad geqip itl

.mixtqenn md (mixai`=) miwiwlgdy xexa dirad geqipn ,ok enk .mj = 1, j ≥ 1 okl,jkl m`zdae Zj ∼ Po( 1

j!), j ≥ 1 ,okae

g(x) =∑n≥0

pn

n!xn = exp

( ∑j≥1

1

j!xj

)=

exp(ex − 1

). (22)

pn = okl .B(n) eze` mipnqne Bell xtqn mi`xew [n] dveawd zewelg ly llek xtqnl:(22) jnq lre B(n)

∑n≥0

B(n)

n!xn = exp

(∑j≥1

1

j!xj

)= exp

(ex − 1

).

19

if`.miwela k-n miakxend πn mipan xtqn B(n, k) onqp

B(n) =n∑

k=1

B(n, k).

.ipy beqn Stirling ixtqn mi`xwp B(n, k):miiwzn :dprh

B(n + 1) =n∑

k=0

(n

k

)B(k).

elceby welaa `vnp n+1 xtqnd l"pd dwelg lka .[n+1] dveawd zewelga opeazp :dgked.mipte`

(nk

)-a xegal ozip wela eze` ly mixtqnd k xzi z` .0 ≤ k ≤ n xy`k ,k + 1

zewelgd xtqnd .mixzepd mixtqn n − k zveaw ly dwelg rval x`yi jkn d`vezk,okl .B(n− k)-l deey dl`d

B(n + 1) =n∑

k=0

(n

k

)B(n− k) =

n∑

k=0

(n

k

)B(k).

.l.y.nzeveaw ax

.`.n mr dpzen xyw i"r mixvep el`d mi`xw`d mipandmixhnxt mr zilily zinepia mibletne mielz izla Zj

:mj, ρj, j ≥ 1

Zj ∼ NBi(mj; ρj), j ≥ 1.

:i"r zxcben zilily zinepia zebltzdy xikfp

a(k)j = P (Zj = k) =

(mj + k − 1

k

)ρjk(1− ρj)mj ,

j ≥ 1, k ≥ 0.

if` ,zeglvd mj lleke cr ilepxa iieqip xtqnk Xj .`.n xicbp m`cr ,mj-l xarn mielz izla ilepxa iieqip xtqnk yxtl ozip Zj okl .Zj = Xj −mj, j ≥ 1

migipn ,z`f xe`l .ρj-l deey ieqip lka dglvdd zexazqd xy`k ,zeglvd mj llekeo`k .0 < ρ < 1-e mj ≥ 1, j ≥ 1-y

a(j)k =

(mj + k − 1

k

)ρjk, j ≥ 1, k ≥ 0

:mexhwtqd xear `ad iehia lawzn (6)-l m`zdae

P(K(n) = η

)= (cn)−1

n∏j=1

(mj + kj − 1

kj

)ρjkj =

20

(cn)−1ρn

n∏j=1

(mj + kj − 1

kj

), η ∈ Ωn,

xy`k

cn = ρn∑η∈Ωn

n∏j=1

(mj + kj − 1

kj

).

:milawn ρn-a mevnvd ixg` ,seqal

P(K(n) = η

)=

∏nj=1

(mj+kj−1

kj

)∑

η∈Ωn

∏nj=1

(mj+kj−1

kj

) . (23)

,ρ-a dielz dppi` mexhwtqd zebltzdy meyn-iwzn .cn dxciq xear g zxvei divwpet dpap dzr .lcend ly iyteg xhnxt `ed oexg`d

:miSj(x) =

∑

k≥0

(mj + k − 1

k

)ρjkxjk =

1(1− (ρx)j

)mj, j ≥ 1.

-y raep jkng(x) =

∏j≥1

1(1− (ρx)j

)mj. (24)

dllkd `id dlawzdy divwpetd( Newton ly mepia zgqep zxfra dyrp oexg`d xarnd).Euleur ly zxvei divwpet ly

.(23) mexhwtqd z` xvei xy` i`xw` dpan zelbl `id d`ad epzxhnlaben `l xtqn qlk`l leki `z lky jk ,mi`z xtqn m-e midf miwiwlg xtqn k idi dnl

-l deey mi`zd oia miwiwlgd ibelit xtqn if`.l"pd miwiwlgd ly(

m + k − 1

k

).

.zihqihhq dwifita gzet dgkedd oeirx:dgkedzenewn m− 1 xgap mkezny zevignd oia zenewn m+k− 1-e zeevwa zevign izy rawp

.(mi`zd oia) zevignl

• • • • • • • •︸ ︷︷ ︸m+k−1

cg` ,`nbecl .mi`z m oia miwiwlg k ly belit mi`zn zevign ly l"pd dxiga lkl:m = 3-e k = 4 xear mibelitdn

• • • •

21

okl.zenewn m+k−1 oia zevign m−1 ly dxiga dni`zn miwiwlgd ly belit lkl ,jtdle`ed miixyt`d mibelitd xtqn

(m + k − 1

m− 1

)=

(m + k − 1

k

)

.l.y.nη = idz .mixtqenn `l xnelk,midf miwiwlg n qlk`nd πn wixt dpana opeazp dzrdpana miwiwlgd xtqn df kj-y jk ,miwixt-i` mipain oia miwiwlgd zwelg (k1, . . . , kn)miwixt-i` mipan=) mi`z mj oia bltl ozip midf miwiwlg n-y epi`x .j lceba wixt-i`

( j lceba-a(

mj + kj − 1

kj

)

l"dq mpyi okl .mipey mipte`

N(η) =n∏

j=1

(mj + kj − 1

kj

)

-ixtd mipand lk ly pn xtqny xne` df.(η dwelg =)mexhwtq eze` ilra miwixt mipan-l deey ,midf miwiwlga miqlk`nd n lceba miw

pn =∑η∈Ωn

n∏j=1

(mj + kj − 1

kj

)= ρ−ncn.

mexhwtqd K(n)-ae l"pd mipand pn lr cig` ote`a bletnd i`xw` dpan Πn-a libxk onqp:miiwzn .xvepy

P (K(n) = η) =N(η)

pn

.

.(23)-l ddf dlawzdy dgqepd

ze`nbec,zeveaw ax md n xtqn ly zellweyn zewelgy d`xn (24) dgqepd .zellweyn zewelg.`miwixt-i` mipan miqlk`n xy` miwiwlgd lcend geqipay dcaerl m`zda oaenk dfe

.midf md:milawn (23)-n.n ly zellweyn `l zewelg −mj = 1, j ≥ 1 dxwna cgeina opeazp

µ(η) = P(K(n) = η

)=

1∑η∈Ωn

1=

1

pn

, η ∈ Ωn.

ieey od n ly zewelg lk ,zexg` milina e` , Ωn lr dcig` zebltzd `id µ-y d`xn df.zexazqd

22

ipya ielz pn,k-e xg`n .cala mixaegn k zeliknd n ly zewelg xtqn pn,k-a dzr onqp:mipzyn ipy ly g(x, z) divwpet `id pn,k dxciqd zxvei divwpet ,mixhnxt

g(x, z) :=∑

k≥0

∑n≥0

pn,kzkxn.

dprh:miiwzn

g(x, z) =∏j≥1

1

1− zxj. (25)

:(25) ly ipnid sb`d ly zewfg xehl gezit `vnp :dgked∏j≥1

1

1− zxj=

∏j≥1

( ∑

l≥0

(zxj)l).

,okl[xnzk]

∏j≥1

1

1− zxj= [xnzk]

∏j≥1

( ∑

l≥0

zlxjl)

=

[xnzk](1 + . . . + zk1+...+knx1k1+2k2...+...nkn + . . .

)= pn,k.

.l.y.nBose− Einstein ly llken lcen :il`ci` fb .a

ly dlewlen lynl ,wiwlg lk ly dibxp`y zraew zihpeew dwipkn ly ceqid zgpd-invr mikxr od l"pd zenxd.cala 1, 2, . . . , miccea (zenxa `vndl=) mikxr lawl dleki,fbmj mpyi dibxp` ly j dnxl=)mj zelitk lra `ed j r"r lky jk ,miieqn xehxte` ly mi-len lynl ,miwiwlgd oia zwlgzn zkxrn ly E = n zllekd dibxp`d jk itl .(mibeqmiaygp miwiwlgdy xg`n .j dnxa dibxp` dpyi miwiwlg kj-n cg` lkly jk ,fb zelewdxwnde Bose−Einstein lcen `xwp mj = 1, j ≥ 1 dxwnd.dveaw ax `ed lcend ,midfk

.Bose− Einstein ly llken lcen `xwp mj = jα, α > 0, j > 0dxwna hxta ,dtiq` `ed fbd lcend if` mixtqenn miaygp miwiwlg m`y xirp seqal

.Bolzman ly lcen dpekn dtiq`d ,mj = jα, j ≥ 1miyxey `lle mixtqenn `l mivr zexri .b

micewcew n lr xri .yxeyk xgadl leki ely eicewcewn cg`y milbrn `ll xiyw sxb epd urmicewcewd .micewcewd n k"dq lr mipapd (miwixt-i` mipan=) mivr dnk ly cegi` `ed

.`l e` mixtqenn zeidl milekio`ky gikedl xyt`.dveaw ax `ed oecipay dxwnd

.c = 0.5349 ,ρ = 0.3383 xy`k ,mj ∼ cρ−jj−52 , j →∞

mixganmiwixt-i` mipann zeakxend zeveaw iaxk mixcben mixgan

mipan ly mibeq mj-n cg` lkn cg` "bivp"n xzei `l "xgap" dpanl xnelk ,efn ef mipeydη = (k1, . . . , kn) ∈ oezp mexhwtq ilra mixgand ly mxtqn df jnq lr .j lceba miwixt-i`

-l deey Ωn

N(η) :=n∏

j=1

(mj

kj

)

23

mixgand mexhwtq ly zebltzdd zgqep z` milawn ep`e:mi`xw`d

P(K(n) = η

)=

∏nj=1

(mj

kj

)

pn

, (26)

xy`kpn =

∑η∈Ωn

N(η).

zinepia mibletnd Zj .`.n mr dpzen xyw "ir zxvep l"pd zebltzddy `ceel xzep seqal, ρj

1+ρj -e mj mixhnxtd mr:mj ≥ 1, ρ > 0 xy`k

P (Zj = k) =

(mj

k

)( ρj

1 + ρj

)k( 1

1 + ρj

)mj−k

,

0 ≤ k ≤ mj.

,mpn`

ajk =

(mj

k

)ρjk,

0 ≤ k ≤ mj.

.(26) dgqep zlawzn jkn d`vezk.lcend ly iyteg xhnxt `ed ρ-y jk ,ρ-a dielz `l mexhwtqd ly (26) zebltzddy oiivp

:lawp jyndaS(j)(x) =

∑

k≥0

ajkx

jk =

mj∑

k=0

(mj

k

)(ρx)jk =

(1 + (ρx)jk

)mj

, j ≥ 1.

-eg(x) =

∏j≥1

(1 + (ρx)jk

)mj

.

`nbecmj = ,n ly zewelg xeary xikfp .mipey mleky mixaegnn zeakxend n xtqn zewelg

.1, j ≥ 1sqe` `ed xgan oecipay dxwna ,jkitl

Ω(d)n := η = (k1, . . . , kn) ∈ Ωn : ki ∈ 0, 1.

,(0, 0, 0, 0, 0, 1) ,(1, 1, 1, 0, 0, 0) ,(1, 0, 0, 0, 1, 0) :od zeixyt`d zewelgd n = 6 xear ,lynl.(0, 1, 0, 1, 0, 0)

.Ω(d)n lr cig` bletn mexhwtmdy d`xn (26)-a mj = 1, j ≥ 1 davdd

.Fermi lcen dpeknd il`ici` fb ly milcenn cg`k ynyn oecipay lcendy oiivp

24

mixgane zeveaw iax,mitqe` oia xywd:mekiqly ze`ad zexveid zeivwpetd yelyl mixeyw l"pd mi`xw`d mipand zyelyy epi`x

cn : zexcqd

:mitqe` •g(x) = exp

(∑j≥1

ajxj), aj =

mj

j!, j ≥ 1

.pn = n!cn, n ≥ 1 :o`k .zikixrn zxvei divwpet -

:zeveaw iax •g(x) =

∏j≥1

(1− (ρx)j)−mj .

:pn, n ≥ 1 dxciqd ly h zxvei divwpet dzr xicbp .pn = ρ−ncn, n ≥ 1 o`k

h(x) =∑n≥0

pnxn.

:miiwzn if`

h(x) =∑n≥0

cn(ρ−1x)n = g(ρ−1x) =∏j≥1

(1− xj)−mj .

.Euleur ly zxvei divwpet z`f

:mixgan •g(x) =

∏j≥1

(1 + (ρx)j)mj .

-y jk ,pn = ρ−ncn, n ≥ 1 aey o`k

h(x) =∑n≥0

pnxn =∑n≥0

cn(ρ−1x)n =

g(ρ−1x) =∏j≥1

(1 + xj)mj .

zeivwpetk meyxl ozip mixgane zeveaw ax xear g zexveid zeivwpetd z` mby dzr d`xponqp .zeikixrn

M(x) =∑j≥1

mjxj.

,zeveaw iax dxwna if`

g(x) = exp(−

∑j≥1

mj log(1− xj))

=

25

exp(∑

j≥1

mj

∑

k≥1

xjk

k

)=

exp∑

k≥1

(M(xk)

k

).

mixgan xear ,ote` eze`a

g(x) = exp∑

k≥1

(M(−xk)

k

).

:lawp jynda

[xl]( ∑

k≥1

M(xk)

k

)= [xl]

( ∑j≥1

mj

∑

k≥1

xjk

k

)=

∑

jk=l

mj

k=: al ≥ 0, l ≥ 1. (27)

:d`ad dxeva meyxl ozip zeveaw iax xear zxvei divwpet jkitl

g(x) = exp( ∑

l≥1

alxl).

mj, j ≥ 1 mixhnxt mr dveaw axy xacd yext.(27) i"r mj, j ≥ 1-n milawznd al, l ≥ 1 mixhnxtd mr dtiq`l dlewy mipezp

(27) dgqepd ik ,cala zilnxet `id l"pd zeliwydy `l`zbdpznd aj dxciql ,j-l qgia (xceqn=)ixlebx ote`a dpzynd mj dxciq lk dxiarn

.(zxceqn `l=)edeae edez zxeva:milawn o`kn .al =

∑jk=l

1k, l ≥ 1 if` .mj = 1, j ≥ 1 idz ,lynl

a1 = 1, a2 = 1 + 1/2 = 3/2, a3 = 1 + 1/3 = 4/3,

a4 = 1 + 1/2 + 1/4 = 7/4, a5 = 6/5, . . .

zexigad iabl .l ly wlgn `ed k-yk 1/k mixtqnd lk ly mekql deey al,illk ote`a,mpn` .izedn ote`a dpey avnd

[xl]( ∑

k≥1

M(−xk)

k

)= [xl]

( ∑j≥1

mj

∑

k≥1

(−x)jk

k

),

.llk zeveaw iaxl zelewy opi` zexiga jkitl.miilily eidi al, l ≥ 1 mincwnd oiay jkKhitchine zhiy zeceqi

oexztl zizexazqd dhy rivd Khitchine 1950-a.zihqihhq dwifita zexxerznd zeihehtniq` zeira

:mi`ad zepexwrd ipy lr zqqazn dhiyd

26

-pet jxc oecipay lcebd z` `hal ick ,iyteg xhnxt mr izexazqd xfr lcen ziipa •ly zebltzdd.mielz izlae micica mi`xw` mipzyn mekq ly zexazqdd zivw

.iyteg xhnxta dielz l"pd .`.n

.iyteg xhnxt ly dni`zn dxiga zxfra ilwel leab htyn zgked •

ligzp .cn znleqn dwelg zivwpet ly dwihehtniq`d xwg xear dhiyd meyi mibcp-y jk ,Zj .`.n i"r zxvepd idylk µn ziaihwiltihlen dcinn

P (Zj = k) = a(j)k , k ≥ 0, j ≥ 1.

:n-a dnehwd gn zxvei divwpet xicbp g zxvei divwpetl sqepa

gn(x) =n∏

j=1

S(j)(x) :=∑

k≥0

ck,nxk, n ≥ 1.

,x ∈ C : |x| ≤ 1 lk xeary oiivp

|S(j)(x)| = |∑

k≥0

a(j)k xjk| ≤ 1

a(j)0

, j ≥ 1.

oigadl lw .l"pd megza qpkzn ok mb gn xehd okl:dfl df mieey gn-e g ly mipey`xd mincwnd (n + 1)-y

.ck,n = cn, 0 ≤ k ≤ n:akexn x dpzynl dzr xearp

x = e−σ+2πiα,

:miiwzne |x| ≤ 1 jkitl .α ∈ R -e σ ≥ 0 xy`k∫ 1

0

gn(x)e−2πiαndα =

∫ 1

0

( ∞∑

k=0

ck,ne−kσ+2πiα(k−n)

)dα = cne−nσ.(28)

zeivwpet zkxrn zeilnxepehxe`a epynzyd oexg`d xarna:e2πiαm, m ≥ 0∫ 1

0

e2πiαmdα =

1, m`, m = 0

0, m`, m 6= 0, m > 0(29)

: σ ≥ 0 iyteg xhnxtl qgia zyweand zedfl miribn ep` (28)-n dzr

cn = enσ

∫ 1

0

gn

(e−σ+2πiα

)e−2πiαndα =

27

enσ

∫ 1

0

n∏j=1

(S(j)

(e−σ+2πiα

) )e−2πiαndα. (30)

-k zedfd z` meyxp ep` (30)-l izexazqd yext wiprdl ick

cn = enσgn

(e−σ

) ∫ 1

0

φ(n)(α)e−2πiαndα. (31)

:i"r zxcben φ(n) divwpet xy`k

φ(n)(α) =n∏

j=1

S(j)(e−σ+2πiα)

S(j)(e−σ).

onqp

pjk =a

(j)k e−σjk

S(j)(e−σ),

-y wiqdl ick

φj(α) :=S(j)(e−σ+2πiα)

S(j)(e−σ), j ≥ 1

:i"r xcbend Xj .`.n ly zipiite` divwpet `id

P (Xj = jk) = pjk, j ≥ 1, k ≥ 0.

mekq ly zipiite` divwpet `id φ(n) =∏n

j=1 φj ,df jnq lr

Vn = X1 + . . . + Xn (31a)

cala minly mikxr lawnd cica Y .`.n xeary xikfp zrk .mielz izla Xj .`.n n ly:miiwzn φY zipiite` divwpet lrae

∫ 1

0

φY (α)e−2πiαndα = P (Y = n).

.zipiite` divwpet ly dxcbde (29)-n raep dfKhintchine zbvd z` (31)-n milawn ep` xac ly enekiqa

:cn znleqn dwelg zivwpet ly .(Khintchine zxeva=)

cn = enσgn(e−σ)P (Vn = n), n ≥ 1. (32)

.lcend ly iyteg xhnxt epid σ ≥ 0 o`kjynda.mpkzn g xehdy jk σ lk xear dyrnl dtwz (32) dbvddy xexa mcewd oeicdn :dxrd

.σ ∈ R xear zniiwzn (32) miniieqn miwixt mipan xeary d`xpdpid ,n → ∞ -yk ,pn lceb edyefi` ly zihehtniq` zebdpzd xwg zxhn llk jxca

.pn ∼ f(n), n →∞-y jk f(n) divwpet z`ivn

28

ilwel leabd htynzexazqdd ly zihehtniq` zebdpzd

P (Vn = n)

leab ihtyn zxez zxfra zrawp (31)-a Vn xear (32)-a.miilwel

zxigal Fawler −Darwin-e Khintchine oexwir (i.).σ xhnxt

σ-a dielz P (Vn = n)-y oiivp ziy`xoecipay zexazqddy jk σ xegal ievx .Xj, j ≥ 1 .`.n ly zeiebltzdd zeielz σ-ay meyn

mixwn daxda (!) zeleki xy` mikxcdn zg`.dlecb didiz-y jk σ = σn xegal `id ,z`f giqadl

EVn = n. (33)

daiaqa zfkexn .`.n ly zizexazqd dqny dcaer lr qqazn z`fd dxigad oei`x:ayiav oeieeiy-i`l m`zda ,ely zlgez ly zniieqn

P (|X − EX| ≥ A) ≤ V arX

A2, A > 0.

:σ-l qgia d`ad d`eeyn zlawzn (33) -nn∑

j=1

∑

k≥0

(kj

a(j)k e−σkj

S(j)(e−σ)

)= n.

.miwixtd mipand zyely xear (33) d`eeynd zxev oldlmitqe`

-y epi`xS(j)(e−σ) = exp (aje

−jσ), j ≥ 1.

,okl

pjk =

(aje

−σj)k

k!exp (−aje

−σj), k ≥ 0, j ≥ 1.

-e xg`npjk := P (Xj = jk) = P (

1

jXj = k),

.`.n1

jXj ∼ Po(aje

−σj), j ≥ 1.

:(33) ly d`ad dxevd z` lawp seqale EXj = jaje−σj, j ≥ 1 jkn d`vezk

EVn := M1(σ) =n∑

j=1

jaje−σj = n, n ≥ 1. (∗)

29

zeveaw iaxo`k

S(j)(e−σ) =1(

1− ρje−σj)mj

, j ≥ 1.

-epjk =

(1− (ρe−σ)j

)mj

(mj + k − 1

k

)(ρe−σ)jk,

k ≥ 0, j ≥ 1.

,mcewd dxwna enk ,jkitl .(ρe−σ)j-e mj mixhnxt mr zilily zinepia zebltzd oini sb`a

1

jEXj =

mj(ρe−σ)j

1− (ρe−σ)j, j ≥ 1

-e

EVn := M2(σ) =n∑

j=1

jmj(ρe−σ)j

1− (ρe−σ)j= n, n ≥ 1. (∗∗)

zexazqd mr miielz izla ilepxa iieqip xtqn `ed X .`.n xy`k,Z = X−m idi :zxekfz.`.n m ly mekqk bivdl ozip X .`.nd z` .zeglvd m lleke cr ,ieqip lka p dglvd

.EZ = mq−m okl .p xhnxt mr cg` lk ,miielz izla miixhne`b

mixgan

S(j)(e−σ) =

mj∑

k=0

(mj

k

)(ρe−σ)jk = (1 + (ρe−σ)j)mj ,

j ≥ 1.

pjk =(1 + (ρe−σ)j

)−mj

(mj

k

)(ρe−σ)jk =

(mj

k

)( (ρe−σ)j

1 + (ρe−σ)j)

)k( 1

1 + (ρe−σ)j

)mj−k

,

k ≥ 0, j ≥ 1.

okl1

jXj ∼ Bi

(mj;

(ρe−σ)j

1 + (ρe−σ)j

), j ≥ 1

-y jkn raep

EXj = jmj(ρe−σ)j

1 + (ρe−σ)j, j ≥ 1

30

-e

EVn := M3(σ) =n∑

j=1

jmj(ρe−σ)j

1 + (ρe−σ)j= n. (∗ ∗ ∗)

.d`ad ce`n daeygd dpekzd zelra (∗), (∗∗), (∗ ∗ ∗) ze`eeyny xazqn.mixzepd mixwnd ipya bj = mj-e mitqe` dxwna bj = aj onql mikqp

dxrdzeveaw iax mexhwtqd zeiebltzd ly (23), (26) ze`gqepn

-ehtniq`d dfilp`a jkitl .ρ xhnxta zeielz `l l"pd zeiebltzddy epi`x dn`zda mixgane.ρ = 1 rawp jynda bivpy zih

,dlyn cigie cg` σn oexzt miiw (∗), (∗∗), (∗ ∗ ∗) ze`eeynd zyelyn zg` lkl dnldxevdn `ed oexztd ,ok lr xzi.oezp n ≥ 1 lk xear

σn = δn + log y, δn → 0, n →∞, y > 0, (34)

mi`ad mi`pzd ipy z` miniiwn bj mixhnxtd m"n`:ε > 0 lk xear

limj→∞

sup(bjy

−jeεj)≥ 1. (35)

lecb witqn n lk xearebn ≤ yneεn. (36)

,σ-l qgia zecxei zeipehepen od Mi, i = 1, 2, 3 zeivwpetd zyely lk:dgked:miiwzn iteq n lk xeary jk

Mi(+∞) = 0, Mi(−∞) = +∞, i = 1, 2, 3.

.l"pd mixwnd zyely lka ,oezp iteq n lk xear σ = σn oexztd zecigie meiw gihan df

.(∗)−(∗∗∗) ze`eeyndn zg` lka cxtpa opeazp ,dnld zprh ly ipyd wlgd z` gikedl ickmeyxp (34)-l witqn (35) i`pzdy dgked myl

σn = δn + log y, y > 0, n ≥ 1.

(∗) :

n =n∑

j=1

jaje−σnj ≥ nane−σnn ≥ nyne−nεe−σnn. ⇒

1 ≥ yne−n(ε+δn) ⇒:lecb witqn n xear

ε + δn ≥ 0, ε > 0. (37)

31

:miiwzn edylk ε > 0-e milecb witqn n1, n xeary raep (36)-n ,jci`n

n = const +n∑

j=n1

jaje−σnj ≤ const +

∞∑j=n1

je(ε−δn)j. ⇒

ε− δn ≥ 0. (38)

.(34)-l miwitqn (36),(35) mi`pzdy epgked okae .δn → 0, n →∞-y raep (38)-e (37)-nif`.(34) miiwzny gipp .dketdd dprhd z` gikep dzr

n =n∑

j=1

jaje−σnj =

n∑j=1

jajy−je−δnj ≤

∞∑j=1

jajy−jeεj,

xnelk ,miiwzn `l (35) m`.ohw witqn ε > 0-e lecb witqn n xear

limj→∞

sup(ajy

−jjeε1j)

< 1

if` ,miieqn ε1 > 0 xear ,

n ≤∞∑

j=1

je−ε1jeεj < ∞,

,(34) miiwzny dgpda ,seqal .dxizq.ε < ε1 lk xear

n ≥ nany−ne−εn,

.l.y.n .ohw ε > 0-e lecb witqn n xear

∗∗,(35)-n if` .ρ = 1 ∗∗-ay xikfp ziy`x

n ≥ nmny−ne−nδn ≥ nyne−εny−ne−nδn ,

mitqe` xear z`fl dnec jxca .ε + δn > 0-y giken df.edylk ε > 0-e lecb witmn n xear.dnld zeprh xzi z` migiken

∗ ∗ ∗:miiwzn ρ = 1 xear o`k

n ≥ nmne−nσn

1 + e−nσn.

,lecb witqn n xear if` ,(34) miiwzn m` okl

1 ≥ mny−ne−nε

1 + e−σnn.

32

-y i`pza ( zqpkzn ziteqpi` dltknd =) zxcben g divwpetd mixgand xeary xikfp

x : |ρx| < 1 ⇔ ρe−σ < 1.

:miiwzi ohw witqn ε > 0-e lecb witqn n xear okl

mn ≤ Ayneεn = ynen(ε+ log An

) = ynenε1 ,

.miiwzn (36)-y d`xn df .izexixy ε > 0-e xg`n ,ε1 = ε + log An

> 0 xy`k.dnld zeprh xzi z` gikedl ozip dnec jxca

(34) miniiwn xy` miwixtd mipand xe`z:dxrd-l lewy (34) i`pzdy oiivp dligz

limn→∞

σn = log y, y > 0

.cgia (35), (36) mi`pzd ipyl lewy oexg`d i`pzd dnld itlyeoldl.l"pd mi`pzd ipy z` miniiwnd Fl miwixtd mipand zwlgn ly dnbec `iap zrk

-y jk miniiw m` aj ³ bj meyxl mikqp

D1 ≤ aj

bj

≤ D2, j ≥ 1.

Fl := bj : bj ³ yjjl−1 logβ j, j →∞, y > 0, l, β ∈ R.

(39)

zeiraa opeazp oldl .bj = mj-yk y ≥ 1 ,(39)-ay raep mj ≥ 1, j ≥ 1 dyixcny oiivl aeygzyelya licadl yi zihilp` dpigany xazqn .cala Fl zewlgn xear zeihehtniq`

:(39)-a l ly jxrl m`zda mi`ad mixwnd

;Logarithmic case l = 0 •;Convergent case l < 0 •.Expansive case l > 0 •

xear Khintchine i"r cer oiiev oldl `iap xy` σn ly yextd .σn zxigal ilwifit weciv.mdylk miwixt mipanl yextd z` lilkp ep` .zihqihhq dwipkn ly miheyt milcen

divwpeta opeazp .(32) Khintchine zbvdl xefgp

Φ(σ) := enσg(e−σ).

divwpetd z` xicbp ok enk

U(σ) := log Φ(σ) = nσ +n∑

j=1

log S(j)(e−σ).

33

-y oiivp(

log S(j)(e−σ))′

σ= −

∑k≥0

(a

(j)k e−σjkjk

)

S(j)(e−σ)= −EXj.

okl-y mi`ex ep` zrk(

U(σn))′

σ= 0,

⇔,df mr cgi .U divwpetd ly zihixw dcewp σn−

(U(σ)

)′′σ

= −n∑

j=1

(EXj

)′σ.

-yxikfp

EXj = −

(S(j)(e−σ)

)′σ

S(j)(e−σ).

:onqp meyixd xeviw mylQ = Q(σ) = S(j)(e−σ)

,if`(EXj

)′σ

= −Q′′

σQ−(Q′

σ

)2

Q2.

o`kQ′

σ(σ) = −∑

k≥0

(a

(j)k jke−σjk

)< 0, σ ≥ 0.

-eQ′′

σ(σ) =∑

k≥0

(a

(j)k (jk)2e−σjk

)> 0, σ ≥ 0.

-y miwitn ep` uxeey -iyew oeieeiy-i`d zxfra dfn(EXj

)′σ

< 0, σ ≥ 0. ⇒(U(σ)

)′′σ

> 0, σ ≥ 0. ⇒.R+-a dly menipin zcewp σn-e R+-a dxerw U divwpet

if` ,ofe`n avna z`vnpd miwiwlg n ly zinpicenxh zkxrnk wixt dpanl qgiizp m`σ-e zkxrnd ly ditexhp`k zyxtzn U divwpetd ,dwinpicenxh ly ipyd wegl m`zdajka `id ditexhp` ly dlecbd zernynd.zkxrnd ly zhlgend dxehxtnhd ly iktedk

34

ly σ = σn l"pd dxiga ok -lr .zkxrnay (zei`xw`=)edeae edezd znx zccen ditexhp`y.zilnipin ditexhp` zlra zkxrn zxvei iyteg xhnxtd

.Vn ly mihp`ixapi`inq (ii.)xfr xneg

m xear E|Y k| < ∞, k ≤ r-y jke φY (α) = Ee2πiαY zipiite` divwpet lra .`.n Y idi:xeliih xehl gezit miiwzn if`.oezp

log φY (α) =r−1∑

k=1

(2πiα)k

k!sk + O(srα

r), α → 0,

EY k, k ≤ r mihpnend jxc mi`hazn mihp`ixeepi`inq mi`xwpd sk mincwnd xy`k:mipey`xd mihp`ixeepi`inqd zyelyd miiehiad `iap Y .`.nd ly

s1 = EY1, s2 = V arY, s3 = E(Y − EY )3, . . . .

mibeqd zyely xear sk, k ≥ 1 mihp`ixeepi`inq xear miyxetn miiehia oldl :dnl:miwixtd mipand

.mitqe` ∗1

jXj ∼ Po(aje

−σj), j ≥ 1. ⇒

φj(α) = exp(− aje

−σj) ∑

k≥0

e2πiαkj (aje−σj)k

k!=

exp(− aje

−σj)

exp(aje

−σje2πiαj)

=

exp(aje

−σj(e2πiαj − 1))⇒

φ(n)(α) = exp( n∑

j=1

aje−σj(e2πiαj − 1)

)⇒

log(φ(n)(α)

)=

n∑j=1

aje−σj(e2πiαj − 1)

:log φ(n) ly xeliih xehl geztn sr, r ≥ 1-d z` `vnl lw zrk(

log(φ(n)(α)

))(r)

α=0= (2πi)rsr, r ≥ 1 ⇒

n∑j=1

jraje−σj(2πi)r ⇒

sr =n∑

j=1

jraje−σj, r ≥ 1.

35

∗∗.zeveaw iax

1

jXj ∼ NeBi(mj; (ρe−σ)j).

.`.n xy`k Xj = X −mj-y dcaera ynzyp

X = Y1 + . . . + Ymj

(ρe−σ)j xhnxtd mr miixhne`b .`.n mj ly mekq `ed,okl .mielz izla (oelykd zexazqd=)

φY1(α) =∑

k≥1

e2πiαk(1− (ρe−σ)j)(e−σ)j(k−1) =

(1− (ρe−σ)j

(ρe−σ)j

)( e2πiα(ρe−σ)j

1− e2πiα(ρe−σ)j

)⇒

φX(α) = e2πiαmj

( 1− (ρe−σ)j

1− e2πiα(ρe−σ)j

)mj ⇒

φXj(α) =

( 1− (ρe−σ)j

1− e2πiαj(ρe−σ)j

)mj ⇒

log(φ(n)(α)

)=

n∑j=1

mj

(log(1− (ρe−σ)j)− log(1− e2πiαj(ρe−σ)j)

).

:log(1− x), |x| < 1 ly xeliih xehl gezita ynzyp

− log(1− e2πiαj(ρe−σ)j) =∑

k≥1

e2πiαjk(ρe−σ)jk

k

(− log(1− e2πiαj(ρe−σ)j)

)(r)

α=0=

mj

∑

k≥1

(2πjk)r(ρe−σ)jk

k= mjj

r(2πi)r∑

k≥1

kr−1(ρe−σ)jk

⇒ sr =n∑

j=1

mjjr∑

k≥1

kr−1(ρe−σ)jk, r ≥ 1.

36

.mixgan ∗ ∗ ∗:mcew dlawzdy zillk dgqepa ynzyp mrtd

φ(n)(α) =n∏

j=1

S(j)(e−σ+2πiα)

S(j)(e−σ), α ∈ R.

,oecipay dxwnaS(j)(e−σ) = (1 + (ρe−σ)j)mj , j ≥ 1 ⇒

log φXj(α) =

mj

(log(1 + (ρe−σ+2πiα)j)− log(1 + (ρe−σ)j)

)

⇒ sr =n∑

j=1

mjjr∑

k≥1

(−1)k+1kr−1(ρe−σ)jk, r ≥ 1.

:miiwzn .dxrdQr(σ) := jr−1

∑

k≥1

(−1)k+1kr−1(ρe−σ)jk =

(−1)(r−1)( ∑

k≥1

(−1)k+1(ρe−σ)jk)(r−1)

σ=

(−1)r−1( (ρe−σ)j

1 + (ρe−σ)j

)(r−1), r ≥ 1. ⇒

Qr(σ) = −(−1)r−1( 1

1 + (ρe−σ)j

)(r−1)> 0, r ≥ 2.

(40)

.miiaeig md miwixtd mipand mibeqd zyely lk ly mihp`ixeep`inqd :dpwqn.Expansive dxwn .sr-eσn ly dwihehtniq` (iii.)

i"r zrawp i`xw` dpan ly dwihehtniq`dy d`xp jyndamipexg`d z` zepkl bedp jk meyn .n → ∞-yk sr = sr(σn), r ≥ 1-eσn ly zebdpzd.dnl .σn = δn + log y-y xikfpe sr = sr(n, σ)-y oiivp .dpan ly miiqiqa mixhnxt-yk miwixtd mipand zyely lk xear ,y ≥ 1-e β = 0 mr (l > 0) Expansive = dxwna

:miiwzn ,σ = σn -en →∞δn ³ n−1/(l+1), sr(n) ³ n(r+l)/(l+1), r ≥ 2.

.sr(n) ≤ sr(n + 1), r ≥ 2 -e δn+1 < δn ,n ≥ N lk xeary jk N ≥ 1 miiw ,dfn dxzixear nδn →∞-e σn > 0 exeary jka oiit`zn Expansive = (l > 0) dxwnd .`:dgked

.miwixt mipan ly mibeqd zyely lk

37

.Euleur ly dnkq zgqep:xfr xnegmireawe n > x0 xear if` .miieqn x0 -n lgd zipehepene dtivx f : R+ ⇒R+ divwpet idz

,Ai, Bi, i = 1, 2 miniieqn

A1 + f(n) +

∫ n

1

f(x)dx ≤n∑

j=1

f(j) ≤

A2 + f(x0) +

∫ n

1

f(x)dx,

-e zcxei f m`A1 + f(x0) +

∫ n

1

f(x)dx ≤

n∑j=1

f(j) ≤ A2 +

∫ n

1

f(x)dx + f(n),

s`ey lxbhpi`d Euleur zgqepa m` :dgqepd ly d`ad dpwqna ynzyp ep` .dler f m`: miiwzn if` ,n →∞ -yk ∞-l

n∑j=1

f(j) ∼∫ n

1

f(x)dx, n →∞.

,ρ = 1-y dgpde Fl, l > 0 zwlgnd zxcbd ,∗∗ -n .zeveaw iaxn ligzp

n =n∑

j=1

jmje−jσn

1− e−jσn≥ D1

n∑j=1

jle−jδn ≥ D1nle−nδn . ⇒

,df jnq lr .δn → 0, n →∞ -y epgked okl mcew.lecb witqn n yk ,δn > 0

n ≥ D1

n∑j=1

jle−jδn ≥ D1e−nδn

n∑j=1

jl, l > 0

- y dcaera epynzyd o`k.n →∞ -yk ,nδn →∞-y mibiyn ep`en∑

j=1

jl ∼∫ n

1

xldx = O(nl+1), l →∞.

,Euleur zgqepe "`"-ay zecaerd zxfra .a

n ≤ D2

n∑j=1

jle−jδn

1− y−je−jδn≤

δ−l−1n D2

∞∑j=1

(jδn)le−jδn

1− e−jδnδn = δ−l−1

n D2

∫ ∞

0

xle−x

1− e−xdx.

38

-y milawn ep` ,meqg lxbhpi`dy meyn,jci`n .reaw D4 > 0 -yk δn ≤ D4n

−1/(l+1), n ≥ 1,

n ≥ D1

n∑j=1

jle−jδn ∼ δ−l−1n D1

∫ ∞

0

xle−xdx.

.δn ³ n−1/(l+1) -y epgked okl .reaw D3 > 0 -yk δn ≥ D3n−1/(l+1), n ≥ 1, ozep df

.mihp`ixeepi`inqd ly dwihehtniq`d z` `vnp.b,sr xear dlawzdy dgqepd itl

sr ³n∑

j=1

jr+l−1

∞∑

k=1

kr−1e−δnjkyj−jk =

∞∑

k=1

kr−1

n∑j=1

jr+l−1e−δnjkyj−jk.

:inipt mekqa Euleur zgqep zxfra lthpn∑

j=1

jr+l−1e−δnjkyj−jk ∼∫ n

1

xr+l−1e−δnxkyx(1−k)dx

= δ−r−ln k−l−r

∫ knδn

kδn

zr+l−1yz

kδn(1−k)e−zdz.

ohw lxbhpi`d ,δn ly zihehtniq` zebdpzd jnq lre y ≥ 1-y dgpda (.z = δnxk davd)okl.k = 1 -yk oeieeiy mr ,k ≥ 1 lk xear Γ(r + l)-l deey e`

n∑j=1

jr+l−1e−δnjkyj−jk ³ δ−r−ln k−l−r ⇒

sr ³ δ−r−ln

∞∑

k=1

k−l−1,

.sr ³ δ−r−ln okl .qpkzn oini sb`a xehd xy`k

xear σn+1 ≥ σn -y gipp .mieqn n -n lgd,n-l qgia zipehepen zcxei σn -y gikedl x`yp.bif` .edylk n

e−jσn+1

1− e−jσn+1≤ e−jσn

1− e−jσn, j ≥ 1,

-y raep (∗∗)-ny jk

n + 1 =n+1∑j=1

jmje−jσn+1

1− e−jσn+1≤

n+1∑j=1

jmje−jσn

1− e−jσn= n +

(n + 1)mn+1y−n−1e−(n+1)δn

1− y−n−1e−(n+1)δn.

39

,n →∞ -yk 0-l s`ey oexg`d xai`d ,okl mcew dlawzdy δn ly dwihehtniq`d llba.δn xear miiwzn xac eze`y xne` df .lecb witqn n xear σn+1 < σn okl .y ≥ 1 lk xear

,miiwzn ,mihp`ixepi`inql rbepa

sr(n + 1) ≥n∑

j=1

mjjr∑

k≥1

kr−1(ρe−σn+1)jk ≥

n∑j=1

mjjr∑

k≥1

kr−1(ρe−σn)jk = sr(n), r ≥ 1,

.n-a zipehepen zcxei σn-y meyn raep oexg`d oeieeiy-i`d xy`kmipand xzi xear dnld zeprh migiken dnec jxca

.mii`xw`d.ilweld leabd htyn zgkede geqip (iv)

.s2 = V arVn := B2n laewny itk onqp oldl

.Vn xear ilweld leabd htynmiiwzn σ = σn xear ,β = 0 mr Expansive dxwna

P (Vn = n) ∼ (2πB2

n

)−1/2, n →∞.

.zeveaw iax xear dgkedd z` ozip :dgkeddxciqd z` xicbp .`

α0(n) := δ(l+2)/2n log n ³ n−(l+2)/2(l+1) log n.

:miiwzn

P (Vn = n) = T = T (n) :=

∫ 1

0

φ(n)(α) e−2πiαndα =

∫ 1/2

−1/2

φ(n)(α) e−2πiαndα, n ≥ 1.

.1-l deey xefgn mr zixefgn φ(n) divwpetdy jk lr jnzqn [−1/2, 1/2] rhwl xarnd o`k,mpn`

φ(n)(α + 1) = Ee2πi(α+1)Vn = Ee2πiαVn ,

ipy ly mekql T lxbhpi`d z` wxtp jynda .cala minly mikxr lawn Vn .`.ny oeikmilxbhpi`

:T2 = T2(n)-e T1 = T1(n)

T1 = T1(n) =

∫ α0(n)

−α0(n)

φ(n)(α) e−2πiαndα,

T2 = T2(n) =

∫ −α0(n)

−1/2

φ(n)(α) e−2πiαndα+

40

∫ 1/2

α0(n)

φ(n)(α) e−2πiαndα.

yk ,T lxbhpi`l dnexzd aexy ze`xdl `id d`ad epizxhn ,ske`d zhiy oexwird itl-y xnelk ,T1 lxbhpi`dn d`a ,n →∞

T2 = o(T1), n →∞.

( ”0” ly ,n → ∞-yk dphwd daiaqd =) [−α0, α0] divxbhpi`d rhwd ly dxigad.a:mi`ad miaeygd mi`pzd ipy meiw dgihan

limn→∞

αrsr = 0, r ≥ 3

-elim

n→∞α2

0B2n = ∞, (41)

zwihehtniq` lr dnl zxfra miiwzn mpn` (41)-y `cep .α ∈ [−α0(n), α0(n)] lk xear:miiqiqad mixhnxtd

|αrsr| ≤ αr0sr ³

(n−(l+2)/2(l+1) log n

)r

n(r+l)/(l+1) =

nl(2−r)2(l+1 logr n →

0, m` r ≥ 3

∞, m` r = 2.

leabd htynd meiwl miwitqn Lyapunov i`pz mi`xwpd (41) mi`pzy d`xp jyndalk xeary raep (41) i`pze σn zxiga ,xelih xehl log φ(n) ly gezitn .ilweld

:miiwzn α ∈ [−α0(n), α0(n)]

log φ(n)(α) =r−1∑

k=1

(2πiα)k

k!sk + O(srα

r) =

2πiαn− 2π2α2B2n + εn, εn → 0, n →∞. ⇒

φ(n)(α)e−2πiαn = exp(

log φ(n)(α)− 2πiαn)∼

exp(− 2π2α2B2

n

), α ∈ [−α0(n), α0(n)]. ⇒

T1(n) ∼∫ α0(n)

−α0(n)

exp(−2π2α2B2

n

)dα =

(2πBn

)−1∫ 2πα0(n)Bn

−2πα0(n)Bn

exp(−u2

2)du ∼

41

(2πBn

)−1∫ ∞

−∞exp(−u2

2)du ∼ (2πB2

n)−1/2.

.n →∞ -yk T2(n) lxbhpi`d xear mqg `vnp dzr .biptn swz `l log φ(n) divwpetd ly mcewd gezitd T2 ly divxbhpi`d megzay oiivp

s`ey `l α l"pd megzay:(ρ = 1 xear ) zedfn ligzp.n →∞-yk 0-l

|φ(n)(α)| =n∏

j=1

∣∣∣∣1− e−jσn

1− e2πiαje−jσn

∣∣∣∣mj

=

exp(−

n∑j=1

mj

2log

∣∣∣∣1− e2πiαje−jσn

1− e−jσn

∣∣∣∣2 )

, α ∈ R.

:miiwzn∣∣∣∣1− e2πiαje−jσn

1− e−jσn

∣∣∣∣2

=

(1− e−jσn cos 2παj

)2+ e−2jσn sin2 2παj(

1− e−jσn)2 =

1 +2e−jσn(1− cos 2παj)(

1− e−jσn)2 = 1 +

4e−jσn sin2(παj)

(1− e−jσn)2 .

:lawp jkn d`vezk|φ(n)(α)| =

exp(−

n∑j=1

mj

2log

(1 +

4e−jσn sin2(παj)

(1− e−jσn)2

)), α ∈ R.

:oeieeiy-i`a xfrplog(1 + x) ≥ x

1 + c, (43)

.edylk reaw c > 0 -e 0 ≤ x ≤ c -ykmr (43) lirtp j ∈ [4σn)−1, n] lk xear

x = xj(n) :=4e−jσn sin2(παj)(

1− e−jσn

)2 ≤ 4e−1/4

(1− e−1/4)2:= c.

if`

|φ(n)(α)| ≤ exp

−

∑

(4σn)−1≤j≤n

C1mje−jσn sin2(παj)

≤

42

exp

−

∑

(4σn)−1≤j≤n

C2jl−1e−jδn sin2(παj)

, α ∈ R,

(42)

.α0 ≤ |α| ≤ 1/2-yk (42)-a opeazp dzrn .y ≥ 1-y meyn raep oexg`d alyd xy`k,lecb witqn n xeary gikedl ozp

Qn(α) :=∑

(4σn)−1≤j≤n

C2jl−1e−jδn sin2(παj) ≥ log2 n,

α ∈ [α0, 1/2]. ⇒ .

T2(n) ≤ exp(− log2 n),

.lecb witqn n xear.l.y.n .T2(n) = o(T1(n))-y dgiken T1(n) lxbhpi`d ly dwihehtniq`d mr d`eydd

onqp .htynd ly izexazqd yext dzr `iap

f(x) =1√

2πV arXexp

(− (x− EX)2

2V arX

), x ∈ R.

:miiwzn hxta .N(EX, V arX) zilnxep zebltzdd zetitv

f(EX) =1√

2πV arX

,Vn .`.n xearEVn = n, V arVn = B2

n.

-k meyxl ozip epgkedy ilweld leabd htyn z`y xne` df

P (Vn = n) ∼ f(n), n →∞.

.ilnxep ilwel leabd htyn z`xwp dlawzdy d`vezd z`f daiqndielz mzebltzdy oeik n-a mielz Xj, j = 1, 2, . . . , n .`.nn cg` lky oiivl aeyg

yleyn jxrn z`xwp .`.n ly l"pd dxciqd df xe`l.σ = σn xhnxta

Xj = Xj(n), j = 1, 2, . . . , n; n = 1, 2, . . . :

X1(1)X1(2) X2(2)X1(3) X2(3) X3(3)X1(4) X2(4) X3(4) X4(4)

43

.cn xear zihehtniq` dgqep (v):`ad ote`a meyxl ozp Expansive dxwn xear (32) Khintchine zbvd z` dzr

cn ∼ (2πB2n)−1/2enσn gn(e−σn), n →∞,

xy`kδn ³ n−1/(l+1), B2

n ³ n(2+l)/(l+1).

,y = 1-yk ,mxear .mitq`a dzr opeazp

gn(x) = exp( n∑

j=1

ajxj), aj =

mj

j!³ jl−1,

l > 0, j ≥ 1 ⇒,Euleur ly dnkq zgqep zxfra

n∑j=1

aje−jσn ³

n∑j=1

jl−1e−jδn ∼ δ−ln Γ(l),

l > 0.

:cn xear dqb zihehtniq` dgqepl liaen df

log cn ³ −1/2 log(2π)− log Bn + nδn + δ−ln Γ(l)

∼ (1 + Γ(l))nl/(l+1), l > 0, n →∞.

-k "jxra",il`ivppetqw` lceb cn-y xne` df

exp(nl/(l+1)

),

.n →∞-yk.(clustering=)zevawzd zira

( zeveaw=) miwixt-i` miaikx oia zwlegn n zllekd "dqnd" mi`xw` miwixt mipanamilcb ipy η ∈ Ωn lk xear xicbp n-d zwelg ly ziqetih dpenz x`zl ick.dpey mlceby

:mi`adqn = qn(η) = max1 ≤ j ≤ n : kj > 0−

-e η ∈ Ωn dpezp dwelga xzeia lecbd aikxd lceb

qn = qn(η) = min1 ≤ j ≤ n : kj > 0−

xear ,dnbecl .η ∈ Ωn dpezp dwelga xzeia ohwd aikxd lceb

η = (0, 0, 3, 3, 0, 1, 1, 0, . . . , 0) ∈ Ω34,

44

miiwznqn = 3, qn = 7.

(xneg=)dqnd zevawzdd zcin z` mipiit`n l"pd milcebd-ay cera ,xnegd ly zilniqwn zexftzdl dni`zn η = (n, 0, . . . , 0)-y xirp .η dwelga

.cala zg` dveawl uawzn xnegd η = (0, 0, . . . , 0, 1).n →∞-yk mzebltzd xewgp ep`..`.nl mikted qn-e qn ,wixt i`xw` dpan ozpda

:miiwznP (qn ≤ r) =

∑η:qn≤r

P(K(n) = η

)=

(cn)−1∑

η:qn≤r

n∏j=1

a(j)kj

, η = (k1, . . . , kn) ∈ Ωn.

,ote` eze`aP (qn ≥ r) =

(cn)−1∑

η:qn≥r

n∏j=1

a(j)kj

, η = (k1, . . . , kn) ∈ Ωn.

:mipniqa ynzyp y`xn mipezp 1 ≤ r, r ≤ n xear ,oldl

c(r)n :=

∑η:qn≤r

n∏j=1

a(j)kj

=∑

η:qn≤r

r∏j=1

a(j)kj

-e

c(r)n :=

∑η:qn≥r

n∏j=r

a(j)kj

.

.a(j)0 = 1, j ≥ 1-y dcaerdn milawzn minekqd jeza xy` zeltknd zeleaby oiivp

.l"pd zeiexazqdd xear Khintchine zbvd gezitn ligzpzexcqd ly zexvei zeivwpet xicbp .c(r)

n , c(r)n xear zebvdd z` lawp lk mcew jk myl

:l"pd

gn(r)(x) =

r∏j=1

S(j)(x) :=∑

k≥0

c(r)k xk, n ≥ 1

-e

gn(r)(x) =

n∏j=r

S(j)(x) :=∑

k≥0

c(r)k xk, n ≥ 1.

idi .(32) ly gezitl dneca lrtp dzr

x = e−σ+2πiα,

45

:miiwzne |x| ≤ 1 jkitl .α ∈ R -e σ ≥ 0 xy`k∫ 1

0

g(r)n (x)e−2πiαndα =

∫ 1

0

( ∞∑

k=0

c(r)k e−kσ+2πiα(k−n)

)dα = c(r)

n e−nσ.

zeivwpetd zkxrn zeilnxepehxe`a epynzyd oexg`d xarna: σ ≥ 0 iyteg xhnxtl qgia zyweand zedfl miribn ep` dzr .(29) d`x ,e2πiαm, m ≥ 0

c(r)n = enσ

∫ 1

0

g(r)n

(e−σ+2πiα

)e−2πiαndα =

enσ

∫ 1

0

r∏j=1

(S(j)

(e−σ+2πiα

) )e−2πiαndα. ⇒

c(r)n = enσgn

(e−σ

) ∫ 1

0

φ(r)(α)e−2πiαndα,

:i"r zxcben φ(r) divwpetd xy`k

φ(r)(α) =r∏

j=1

S(j)(e−σ+2πiα)

S(j)(e−σ), α ∈ R.

onqp

pjk =a

(j)k e−σjk

S(j)(e−σ),

-y wiqdl ick

φj(α) :=S(j)(e−σ+2πiα)

S(j)(e−σ), j ≥ 1

:i"r xcbend Xj .`.n ly zipiite` divwpet `id

P (Xj = jk) = pjk, j ≥ 1, k ≥ 0.

mekq ly zipiite` divwpet `id φ(r) =∏r

j=1 φj ,df jnq lr

V rn = X1 + . . . + Xr

.mielz izla Xj .`.n r lyz` milawn ep` xac ly enekiqa .σ = σn-y meyn `a n qwcpi`d oldle o`ky oiivl yi

:c(r)n ly Khintchine zbvd

c(r)n = enσg(r)

n (e−σ)P (V (r)n = n), n ≥ 1. (44)

46

:c(r)n ly Khintchine zbvd mb zlawzn jxc dze`a

c(r)n = enσg(r)

n (e−σ)P (V (r)n = n), n ≥ 1. (45)

.Expansive dxwna mitq` xear rvap oecipay zeiexazqdd ly zihehtniq`d dfilp`d z`-k meyxl ozip (44), (45) z` mitq` xear

c(r)n = enσ exp (

r∑j=1

aje−jσ)P (V (r)

n = n)

-e

c(r)n = enσ exp (

n∑j=r

aje−jσ)P (V (r)

n = n),

xy`k

φ(r)(α) = exp( r∑

j=1

aje−σj(e2πiαj − 1)

)

-e

φ(r)(α) = exp( n∑

j=r

aje−σj(e2πiαj − 1)

).

:i"r mipezp Vn.`.nd mihp`ix`eepi`inqdy epi`x

sk =n∑

j=1

jraje−σj, k ≥ 1.

:md V(r)n -e V

(r)n .`.nd mihp`ix`eepi`inqd ,jkitl

s(r)k =

r∑j=1

jkaje−σj, s

(r)k =

n∑j=r

jkaje−σj.

:mcewd oexwrd itl dyrp σ iyteg xhnxtd zxiga z`

E(V (r)n ) =

(r)∑j=1

jaje−σj = n (46)

-e

E(V (r)n ) =

n∑j=r

jaje−σj = n, n ≥ 1 (47).

.dn`zda σ(r)n -e σ

(r)n ,cigi `ede oexzit miiw l"pd ze`eeynn zg` lkly gikedl lw

-ixtd mi`xw`d mipand Fl zwlgn ly d`ad dveaw zz xear gqpp ze`ad ze`vezd z`:miw

F ′l = aj ∼ jl−1, l > 0.

47

.0 ≤ β, β ≤ 1 xy`k r = nβ-e r = nβ eidi :dnlif` ,0 ≤ β < (l + 1)−1 -e (l + 1)−1 < β ≤ 1 m` (i)

σ(r)n ∼ σ(r)

n ∼(Γ(l + 1)

) 1l+1

n−1

l+1 , l > 0, n →∞.

if` ,(l + 1)−1 < β < 1 -e 0 < β < (l + 1)−1 m` (ii)

σ(r)n ∼ − γ log n

nβ(1 + δn), l > 0, n →∞,

-ykγ = 1− (l + 1)β, δn =

log(γ log n)

γ log n,

-eσ(r)

n ∼ γ log n

nβ(1− δn), l > 0, n →∞,

-ykγ = (l + 1)β − 1, δn =

log(γ log n)

γ log n.

if` ,β = β = (l + 1)−1 m` (iii)

σ(r)n ∼ An−

1l+1 , l > 0, n →∞

-eσ(r)

n ∼ An−1

l+1 , l > 0, n →∞,

ze`eeynd ly micigi zepexztk mixcben A, A > 0 -yk

Al+1 =

∫ A

0

tle−tdt

-eAl+1 =

∫ ∞

A

tle−tdt

.dn`zda:dgked

.`nb zivwpet :rwx xneg .`divwpetd z` aigxdl dzid dzxhny 1729 zpyn Euleur ly ez`vnd `id Γ divwpetd

mixtqnl miirah mixtqnn n!:i"r zxcben Γ divwpetd.miakexn

Γ(x) = limk→∞

k!kx

(x + 1)k

,

48

x ∈ C, x 6= −n, n ≥ 0,

xy`k(x)k := x(x + 1) . . . (x + k − 1), k ≥ 1, (x)0 = 1.

.x ∈ C, x 6= −n, n ≥ 0 megza miiw l"pd leabdy gked,l"pd megzay `ceel lw

Γ(x + 1) = xΓ(x).

,mpn`

Γ(x + 1) = limk→∞

k!kx+1

(x + 2)k

=

limk→∞

k!kx

(x + 1)k

k(x + 1)

x + k= (x + 1)Γ(x).

Γ divwpetd Re(x) > 0 megzay `id daeyg dcaerd .Γ(n) = n!, n ≥ 1 -y raep dfn:zilxbhpi` dbvd i"r zpzip

Γ(x) =

∫ ∞

0

tx−1e−tdt, Re(x) > 0.

Re(x) ≤ -yk xcazne Re(x) > 0-yk miiw lxbhpi`dy jk ,tx−1e−t ∼ tx−1, t → 0 -y oiivpmiahew ody −n, n ≥ 0 zecewp hrnl C xeyin lkl zihilp` dkynd dpyi l"pd dbvdl .0

.Γ divwpetd ly miheyt-epdn zg` lky `cel witqn ,l"pd ze`eeyndn zg` lk ly oexztd zecigi llba.aeply ilkd ,mcew enk.dni`znd d`eeynd z` zniiwn (i) − (iii) zeihehtniq` ze`gq-petd xear dgqepd z` lirtp ziy`x.Euleur ly dnkq zgqep `ed zihehtniq` dfilp`l,zihehtniq` dxeva (47) -e(46) ze`eeynd z` meyxl ick f(x, σ) = xle−σx, l > 0 divw

.n →∞-yk:lawzn uσ = x davd ixg`

n ∼(|σ(r)

n |)−(l+1)

∫ r|σ(r)n |

|σ(r)n |

tl exp(− tsign(σ(r)

n ))dt

-e

n ∼(|σ(r)

n |)−(l+1)

∫ n|σ(r)n |

r|σ(r)n |

tl exp(− tsgn(σ(r)

n

)dt.

(48)

l"pd ze`gqepay milxbhpi`d I(n), I(n)-a xeviw myl onqp jyndamiiwzn (i.)

|σ(r)n | → ∞, n →∞

-e|rσ(r)

n | → 0, n →∞. ⇒

49

I(n), I(n) → Γ(l + 1), l > 0,

ikσ(r)

n , σ(r)n > 0. ⇒

(|σ(r)

n |)−(l+1)

I(n) ∼ n

-e(|σ(r)

n |)−(l+1)

I(n) ∼ n.

l.y.nokl .γ, γ > 0 o`k (ii).

σ(r)n < 0, σ(r)

n > 0.

jk ,witqn lecb `l mekqay mixaegnd xtqn" oecipay dxwnay jkn zraep σ(r)n < 0

,lecb n-yk ,n-l deey didi mekqdy icky."1-n dlecb idiz dhppetqw`dy yexc

:miiwzn|rσ(r)

n | = γ(1 + δn) log n →∞, n →∞-e

|rσ(r)n | = γ(1 + δn) log n →∞, n →∞. ⇒

I(n) →∞,

ikσ(r)

n < 0.

:miiwzn .lhitel llka ynzyp jk myl .I(n) lxbhpi`d ly dwihehtniq`d z` `vnp zrk(|rσn|

)′n

= γ(

log n +log(γ log n)

γ

)′n

=

γn−1(1 +

1

γ log n

).

(I(n)

)′n∼

(|rσn|

)l

e|rσn|(|rσn|

)′n, (48)

xy`ke|rσn| = nγ(γ log n).

-k meyxl ozp (48) z` okl(I(n)

)′n∼ (γ log n)l+1(1 + δn)lnγ−1γ

(1 +

1

γ log n

)∼

50

(γ log n)l+1n−(l+1)β, n →∞.

-y xne` df ,lhitel llk itl

I(n) ∼ (γ log n)l+1n−(l+1)β+1 = (γ log n)l+1nγ ⇒

(|σ(r)

n |)−(l+1)

I(n) ∼ n, n →∞.

.dnec r xear dgkegd:miiwzn ,oecipay dxwna (iii)

σ(r)n , σ(r)

n > 0,

rσ(r)n → A, rσ(r)

n → A, nσ(r)n →∞, n →∞.

:dn`zda lawp (48) ly ipni sb`a zepexztd ly mzavd ixg` okl

(|σ(r)

n |)−(l+1)

I(n) ∼ A−(l+1)n

∫ A

0

tle−tdt = n

-e(|σ(r)

n |)−(l+1)

I(n) ∼ A−(l+1)n

∫ ∞

A

tle−tdt = n

zepexzt md A, A-y `cel wx x`yp .A, A ly zexcbdn raep oexg`d alyd mixwnd ipyaivwpetd .dnld zprhay z`eeynd ly micigi miiaeig

F (A) :=

∫ A

0

tle−tdt → Γ(l + 1), A →∞, F (0) = 0.

eze`.cala zg` dcewpa miybtp A(l+1)-e F (A) zeivwpetd izy ly mitxbdy raep dfn.l.y.n .A xear mb swz wenip

.s3-e B2 mixhnxtd dwihehtniq` :dpwqn,(n -a ielz `l xnelk) reaw h > 0-e n →∞-yk f` .dnl itl mipezp σ

(r)n , σ

(r)n eidi

B2 ∼ h

n(σ

(r)n

)−1

, `m (l + 1)−1 ≤ β ≤ 1

nr, `m 0 < β < (l + 1)−1,

B2 ∼ h

n(σ

(r)n

)−1

, `m 0 ≤ β ≤ (l + 1)−1

nr, `m (l + 1)−1 < β < 1.

-es3 ∼ h

(B2)2

n, s3 ∼ h

(B2)2

n, n →∞.

51

,miiwzn :dgked

B2n = s2 ∼

r∑j=1

jl+1e−jσ(r)n

∼(|σ(r)

n |)−(l+2)

∫ r|σ(r)n |

|σ(r)n |

tl+1 exp(− tsign(σ(r)

n ))dt.

(49)

,(49)-n .(l + 1)−1 ≤ β ≤ 1 dxwna opeazp dligz

B2n ∼

(σ(r)

n

)−(l+2)∫ h1

0

tl+1 exp(−t)dt,

-yk

h1 =

(Γ(l + 1)

) 1l+1

, `m β = (l + 1)−1

∞, `m β > (l + 1)−1.

,jkn d`vezkB2

n ∼ hn(σ(r)

n

)−1

.

.dpwqnd zeprhd x`y zelawzn dnec ote`a.dft xarne zeihixw zecewp:dxrd

ly zihehtniq`d zebdpzddy d`xn dnld ze`vez gezip1

l+1, l > dcewp jxc xaer β -yk izedn ote`a dpzyn r = nβ ly divwpetk σ xhnxtd

.lecb n xear ,σn = σn(β) divwpetd ly dvitw zcewp `id β = 1l+1

,zexg` milina,0zeihixw zecewp mi`xew df beqn zecewpl rah ircna

.lcen ly (= Critical Points).(= Phase Transition) dft xarn ygxzn lcenay xnel bedp zihqihhq dwifita ,hxta

zpenz lr l"pd zihixwd dcewpd dzrtyd d`xp jynda.lcend i"r zexvepd zewelgd zevawzdd

.ilwel leab htyn:miiwzn if` .dnla enk σ

(r)n , σ

(r)n eidi

P (V (r)n = n) ∼ (2πB2

n)−12 , n →∞,

P (V (r)n = n) ∼ (2πB2

n)−12 , n →∞.

-y epi`x:dgkedP (V (r)

n = n) =

∫ 1

0

φ(r)(α)e−2πiαndα,

52

i"r dpezp mitq` ly dxwna φ(r) zipiite` divwpetd xy`k

φ(r)(α) = exp( r∑

j=1

aje−jσ

(r)n (e2πiαj − 1)

), α ∈ R.

,1-l deey xefgn mr zixefgn φ(r) divwpetdy jk lr jnzqda

P (V (r)n = n) =

∫ 1/2

−1/2

φ(r)(α) e−2πiαndα :=

T1(n) + T2(n), n ≥ 1,

epniq ep` xy`k

T1 = T1(n) =

∫ α0(n)

−α0(n)

φ(r)(α) e−2πiαndα,

T2 = T2(n) =

∫ −α0(n)

−1/2

φ(r)(α) e−2πiαndα+

∫ 1/2

α0(n)

φ(r)(α) e−2πiαndα.

:aepetl i`pz miiwzn ,dnln dpwqnd itl

s3

B3→ 0, n →∞.

meyxp dzrα0s3 = (α0B)3 s3

B3

-y jk α0 = α0(n) miiwy dpwqn wiqdl ick

limn→∞

α0B = +∞

limn→∞

α30s3 = 0 ⇒

T1 ∼∫ α0

−α0

exp(−2π2α2B2

)dα =

1

2πB

∫ 2πα0B

−2πα0B

exp(−z2

2)dz ∼ 1√

2πB2, n →∞.

:miiwzn.T2 = o(T1), n →∞ -y dgked jxc oiivp zrk

|T2| = 2|∫ 1/2

α0

ϕ(r)(α)e−2πiαndα|.

53

|ϕ(r)(α)| = exp

(−2

r∑j=1

aje−jσ

(r)n sin2 παj

), α ∈ R.

V (r)n (α) = 2

r∑j=1

aje−jσ

(r)n sin2 παj, α0 ≤ α ≤ 1/2.

-y `cel lw

α20 =

log4(B2)

B2,

,l"pd α0-d zxiga xeary gikedl ozip .aepetl i`pz miiwn

e−V(r)n (α) = o(B−1), α0 ≤ α ≤ 1/2, n →∞. ⇒

T2 = o(T1), n →∞.

.l.y.n .dnec jxca migiken P (V(r)n = n) xear htynd zprh z`

.c(r)n , c

(r)n xear zeihehtniq` ze`gqep :dpwqn

if` .r = nβ, 0 ≤ β ≤ 1-e r = nβ, 0 < β ≤ 1 eidi

c(r)n ∼ (2πB2)−

12 exp

(S(r)

n (e−σ(r)n ) + nσ(r)

n

), n →∞,

(50),

c(r)n ∼ (2πB2)−

12 exp

(S(r)

n (e−σ(r)n ) + nσ(r)

n

), n →∞,

(51)

xy`kS(r)

n (e−σ(r)n ) ∼ h

n2

B2, (52)

S(r)n (e−σ

(r)n ) ∼ h

n2

B2, n →∞, (53).

cera ,Khintchine zebvdne ilweld leabd htynn zelawzn (50), (51) ze`gqepd:dgkedzxfra zelawzn (52), (53)-y

,mpn`.Euleur ly dnkq zgqepS(r)

n (e−σ(r)n ) ∼

(|σ(r)

n |)−l

∫ r|σ(r)n |

|σ(r)n |

tl−1 exp(− tsign(σ(r)

n ))dt :=

54

(|σ(r)

n |)−l

In(l − 1)

-eS(r)

n (e−σ(r)n ) ∼

(|σ(r)

n |)−l

∫ n|σ(r)n |

r|σ(r)n |

tl−1 exp(− tsgn(σ(r)

n

)dt :=

(|σ(r)

n |)−l

In(l − 1).

-y raep dfnS

(r)n (e−σ

(r)n )

n∼ σ(r)

n

In(l − 1)

In(l)

-eS

(r)n (e−σ

(r)n )

n∼ σ(r)

n

In(l − 1)

In(l).

,dfd dxwna ,dnl itl .0 < β < (l + 1)−1 dxwna opeazp

σ(r)n < 0, |rσ(r)

n | → ∞, n →∞.

,oklIn(l − 1)

In(l)∼ |rσ(r)

n |l−1

|rσ(r)n |l

⇒

S(r)n (e−σ

(r)n )

n∼ 1

r⇒

S(r)n (e−σ

(r)n ) ∼ n2

nr∼ h

n2

B2,

x`ya dprhd z` migiken jxc dze`a .dnln dpwqnn raep oexg`d xarnd xy`k.mixwnd

zeiexazqdd zwihehtiq` xwgl zybl zrk xyt` zencewd ze`vezd zxfra

d(r)n := P (qn ≤ r) =

c(r)n

cn

-e

d(r)n := P (qn ≥ r) =

c(r)n

cn

.

55

.dqnd zevawzdd lr htyn:miiwzn Expansive mitq` ly dxwna

limn→∞

d(r)n =

0, `m 0 ≤ β ≤ (l + 1)−1

1, `m (l + 1)−1 < β ≤ 1.

limn→∞

d(r)n =

0, `m r = nβ, 0 < β ≤ 1

exp(−∑r−1

j=1 aj

), `m r ≥ 2 nqtx qeti.

xicbpe .cn = c(n)n -y oiivp :dgked

∆(r)n := S(r)

n

(e−σ

(r)n

)− S(n)

n

(e−σ

(n)n

)+ n

(σ(r)

n − σ(n)n

).

-y ze`xdl `id dpey`xd epizxhn

limn→∞

∆(r)n =

−∞, `m 0 < β ≤ (l + 1)−1

0, `m (l + 1)−1 < β ≤ 1.(54).

dxwna ,mpn`.zncewd zihehtniq`d dfilp`dn zexiyi raep (54) ly oey`xd wlgd,dfd

S(r)n (e−σ

(r)n ) ∼ h

n2

B2∼ h1n

1−β, 0 < β ≤ (l + 1)−1

-enσ(r)

n ∼ n(− γ log n

nβ(1 + δn)

)∼

−hn1−β log n → −∞, n →∞,

-y cera .0 < β < 1l+1

xy`k

S(n)n

(e−σ

(n)n

)∼ h

n2

B2∼ h

n2

n(σ

(n)n

)−1 ∼ h2n1− 1

l+1

-enσ(n)

n ∼ hn1− 1l+1 .

:milawn jkn d`vezk

∆(r)n ∼ −hn1−β log n → −∞, n →∞,

56

-y `ed cgein xac ,β = 1l+1

dxwna .0 < β < 1l+1

xy`k

σ(r)n ∼ An−

1l+1 , σ(n)

n ∼ hn−1

l+1 ,

-reawd z` oeayga zgwl jixv n(σ

(r)n − σ

(n)n

)ly zihehtniq` zebdpzdd zelbl icky jk

,dnl itl .h, A mi

Al+1 =

∫ A

0

tle−tdt < Γ(l + 1).

,okl

A < h =(Γ(l + 1)

) 1l+1 ⇒

nσ(r)n − nσ(n)

n → −∞, n →∞.

mby migiken ote` eze`a

S(r)n

(e−σ

(r)n

)− S(n)

n

(e−σ

(n)n

)→ −∞.

-y 0 < β ≤ 1l+1

dxwnay eplaiw dgkedd jldnay oiivl aeyg

∆(r)n ≤ −nγ,

dcaerd lya ,limn→∞ d(r)n xear htynd zprh ly oey`x wlg raep dfn .miieqn γ > 0 mr

.B(r)n ∼ hnγ, γ > 0-y

,(54) ly ipyd wlgl mi`znd dxwna

σ(n)n ∼ σ(r)

n , n →∞,

jk meyneS(r)

n

(e−σ

(r)n

)∼ S(n)

n

(e−σ

(n)n

), n →∞.

oeieeiydn .dpicr xzei dfilp`a jxev yi o`k jk meyn

r∑j=1

jaje−σ

(r)jn −

n∑j=1

jaje−σ

(n)jn = 0, n ≥ 1

-y raep dnlne-ye σ

(n)n > σ

(r)n ≥ 0,

n(σ(n)n − σ(r)

n ) → 0, n →∞.

oecipay dxwna,mpn`

57

(σ(r)

n

)−(l+1)∫ rσ

(r)n

σ(r)n

tl exp(−t)dt =

(σ(r)

n

)−(l+1)

(Γ(l + 1)− ε1(n)),

xy`k

ε1(n) = Γ(l + 1)−∫ rσ

(r)n

σ(r)n

tl exp(−t)dt ∼

exp(−rσ(r)

n

)(rσ(r)

n

)l

, n →∞,

,jk meyn .lhitel llk itlnpε1(n) → 0, n →∞,

p lk xear⇒

nσ(r)n = (Γ(l + 1))

1l+1 n

ll+1 + vn,1, vn,1 → 0, n →∞.

,ote` eze`ae

nσ(n)n = (Γ(l + 1))

1l+1 n

ll+1 + vn,2, vn,2 → 0, n →∞.

-y migiken miwenip mze` zxfra .dprhd z` giken df

S(r)n

(e−σ

(r)n

)− S(n)

n

(e−σ

(n)n

)→ 0, n →∞.

dgkegd jxc x`zp dzr .d(r)n iabl htynd ly dipyd dprhd zlawzn xac ly enekiqa

.d(r)n iabl htynd zprh

if` .edylk oezp iteq xtqn r ≥ 2 idi

σ(r)n ∼ σ(1)

n , n →∞,

limn→∞

n(σ(r)

n − σ(1)n

)= 0.

meyxp jyndaS(r)

n

(e−σ

(r)n

)− S(1)

n

(e−σ

(1)n

)=

n∑j=1

aje−σ

(1)n j

(e−(σ

(r)n −σ

(1)n )j − 1

)−

r−1∑j=1

aje−σ

(r)n j.

(55)

58

-e σ(1)n > σ

(r)n ≥ 0 ,miiwzn

e−(σ(1)n −σ

(r)n )j − 1 = −

(σ(1)

n − σ(r)n

)j(1− δn),

,jk meyn .1 ≤ j ≤ n lkl qgia cig` ote`a ,δn = δn(j) → 0, n →∞ xy`k ,n∑

j=1

aje−σ

(1)n j

(e−(σ

(r)n −σ

(1)n )j − 1

)=

−((σ(1)

n − σ(r)n )(1− δn)

) n∑j=1

jaje−σ

(1)n j =

−((σ(1)

n − σ(r)n )(1− δn)

)n → 0, n →∞.

-l s`ey mixai` ly iteq xtqn mekq `edy ipyd mekqdy cerar−1∑j=1

aj

.l.y.n .l > 0 ,mitq`:zevawzdd zpenz

.σ(r)n , σ

(r)n ly zebdpzd •

zeivwpetk ze- σ ly zebdpzda opeazpe lecb witqn n z` `itwp.r = nβ, r = nβ eidi-y epi`x .cala β ly

1

l + 1

.β xhnxtd ly zihixw dcewp `id.htynd ze`vez gezip

zeiexazqdd zebdpzday d`xn htynd

d(r)n := P (qn ≤ r)

-ed(r)

n := P (qn ≥ r)

dcewp dze`a β xhnxtl qgia dft xarn ygxzn ok enk1

l + 1

lecb witqn n xeary raep htynn .zxcbd itl ,0-l deey n

1l+1 lceba (dveaw=) aikx didi n ly dwelgay zexazqdd (i)

;zebltzd divwpetPo(aj), j = mibletne miielz izla md K

(n)1 , . . . , K

(n)p .`.nd ,oezp iteq p lk xear (ii)

.dn`zda, 1, . . . , p

.dveawd lceb xear sq=Threshold `xwp n1

l+1 xtqnd ,df xe`l

59