On limit distribution of sums of random variables Текст научной статьи по специальности «Математика»

УДК 519.21

On Limit Distribution of Sums of Random Variables

Sergey V. Chebotarev*

Altai state pedagogical university Molodezhnaya, 55, Barnaul, 656015

Russia

Received 24.06.2015, received in revised form 29.12.2015, accepted 12.01.2016 Centered Rademacher sequences and centered sequences of lattice random variables with a non-trivial 1 "

weak limit of the sums > are considered in the article. A general form of limit distribution is

Vn ^ v i=i

found for these sequences. It is shown that the form of limit distribution depends only on the average mixed moments of the first order characterizing random variables of the sequence. In the case of lattice random variables we mean a sequence of Rademacher random variables in which we can distribute the elements of the given sequence.

Keywords: sequences of random variables, sum of random variables, sum of dependent random variables, limit distribution.

DOI: 10.17516/1997-1397-2016-9-1-17-29.

Introduction

Features of limit distribution of sums of random variables are some of the most actively discussed problems [1]. A rather detailed study of the features of limit distribution of sums of random variables is presented in [2]. It provides a general view on the limit distribution for Rademacher and lattice random variables. Characteristic studies in this field are given in [3-6].

We study the sequences of random variables £ = (^t)tei, defined on probability space (QI, AI, P^), where I is some set of indexes. It is assumed that random variables are defined on similar spaces of elementary events Qt = Q,t G I, and QI = Q1 x Q2 x ■ ■ ■ x Qt x ... with similar algebras of events At = A and AI = A1 x A2 x ... x At x .... The values of random variables lie in the value space £t(w) G Xt, w G Qt. Two kinds of sets are used as value spaces of random variables. In the case of Rademacher random variables X(0) = [—0; 0], that is a set which consists of two numbers 0 and -0. We assume that P(£t = 0) = pt, and P(£t = -0) = 1 — pt = qt. In the case of lattice random variables, it is a set consisting of s + 1 numbers X(0, s) = [0(2k — s); k = 0,1,..., s], where 0 is a step of lattice distribution and

s

P(£t = 0(2k — s)) = pt(k); k = 0,1,..., s; ^ pt(k) = 1. There are no additional limits for the

k=0

values of combined probability of sequence elements £. The main result presented in the paper is Theorem 3.

1. Preliminary results

Let us consider sequences of random variables £ = (£t)teN, N = [1,2,...], |E£t| < to, where

1 n

- V E£t -► 0

n z—' n^rco

t=1

* svcheb@gmail.com © Siberian Federal University. All rights reserved

takes place. The finite sequences of random variables are denoted by

Si = (St)tei, I = {t!,...,tm}.

In the case when I = Im = {1, 2,..., m} we write either (Zt)teIm or S(m).

Let us consider the subsequences of the finite sequence S(n). Let I = {t1,... ,tm} C In and \I\ is the cardinality of the set I. The initial mixed moment m of random variables £tl,... ,£tm of order \I\ = m is denoted by vI:

VI = mK,,A (St, ,...,Ztm ) = mt> ■■■ (Stm )'.

Let us introduce the total mixed moment of order m as

vm vi, ym = 1,... ,n, when m = 0,vo = 1.

\I\=m

Two kinds of average moments of order m are used:

vm = , ym = 1,... ,n; m rm^ ii)

Cn

and

v

ym = 1, ... ,n.

Here Cnm is a binominal coefficient. It is the number of combinations from n by m. When total mixed moment vm or some average mixed moments vm, vm are defined for the sequence S we write vm(S) or vvm(S), ^m(S), respectively. Let us introduce a random variable

1n

Sa (S(n)) = na^ St.

t=l

It can be also represented as

n

S0(n(n)) =^2 nt,n, t=1

where ntn = 0nSt, and 0n = —. It is shown in theorem 2.2 [7] that in the case when the sequence

na

of Rademacher random variables with the values nt(w) e {-9,0} = X(0), w € Q,t, t e In is defined then the following relations are satisfied

C k

P(S0(n{n)) = e(2k - n))= Pn(„, = d-m*m(n(n)) • Bn(m,k) (1)

m=0

and

vm(n(n)) = 0mY, P-M (k) ■ Bn(m,k) ym > 1, (2)

m=0

where

m

Bn(m, k) = -1)mj2 (-1)ick ■ cm-i i=0

Let us note that there is a correlation between the total mixed moment vm(S(n)) and the total mixed moment vm(n(n)):

vm(n(n)) = vi (n(n)) = 9m vi (S(n)) = 9mvm(S(n)) ym > 1,

\I\=m \I\=m

and when m = 0 we assume v0(nçn)) = v0(£çn)) = 1. When m > 1 we have similar relations for

= = emim(£{n)) and vm(n{n)) = = ^^П))- (3)

Let n be some absolutely continuous random variable with probability density function . We denote by Lm(n) the value of integral

1 rtt rtt

Lm(n) = Hm(x)^n(x) dx = hm(x)^n(x) dx; m G N.

V m\ J J-tt,

We assume that it exists and has a finite value. Here Hm is the orthogonal and hm is the orthonormal Hermite polynomial of degree m. We also introduce

Л,

following formalism: hl(mn) means that all terms xk in the polynomial hi(x) are replaced by

mk(n) = x (x) dx.

J —ж

It is the initial moment of degree к of the random variable q. At the same time we use the lowing formalism: hi I mk(n), к = 0, 1, .. . ,m.

Let us consider the sequence of Rademacher random variables (£t)teN, £t(^) G Xt(0 = 1),ш e 1 n

Qt, E£t = 0. Sums ^^Y^ £i converge weakly to an absolutly continuous random variable n with

Vn i=i

probability density function . Then

Theorem 1. For any m so that Lm(n) exists and has a finite value the following relations are true:

vm(£(n)) 1 , ,

гш = Lm(n). (4)

п^ж Jnm \Jm\

lim ijm(£(n))Vnm = \fm\ • Lm(n). (5)

lim vm(£(n)) = Lm(n). (6)

lim vm(£(n)) = 0. (7)

Proof. To prove (4) we use relation (2) and the results given in [8]. It follows that Bn(m,k) with fixed n are the Kravchuk orthogonal polynomials [9] (p = q = ^) with respect to integer variable k. They are asymptotically related to the Hermite polynomi. For any converging sequence on the extended domain of real numbers

xn — /— ^ x

and a corresponding

n

sequence (kn), kn G [0,1, 2,..., n] the following statement is true uniformly in relation to x:

Bn(m,kn) 1 . . 1 ,

lim -— = — Hm(x) = hm(x).

„ ____/ >rr-1 „„.j I v ' / t v '

и^ж y/nm m! ym!

For a big value n we have

2k — n

Pn(n) (k) = Pn(n) (Хи) « Un (xu)Axu, where Хи = —.

Then we obtain

Bn(m,k) exchange

vm(n(n)) = Y^ (k)

n(n> *

k=0

¿i °(—n)) • P->"- )

= v n \ /

1 ( ^ mil ^

(xn (xn)

Xn= — -Jn

Thus

( Tn ))

xn + Oi( ^ ) ) ^ —= Lm(n).

Jmi

It proves relation (4). To prove (5) we use the relation

vm(n(n)) = 9mvm(S(n)) = ^^ ^ Lm(n)- (8)

nm m!

nm

It is true for any fixed m. Taking into account that li^ -r = 1, for a big value n we

n^w Cm1 m!

have vm(S(n)) = Cmv>m(S(n)) and

nm

vm (S(n) ) ~ m vm(S(n)).

The proof of (6) is the same and (7) follows from the previous relations. □

Let us assume that the average mixed moments of the sequence S exist. Then their limits are

-nmvm(S(n)) ~ -nmm ■ vm(S(n)) = m! —Sm)') ^ —mm!Lm(n).

nm nm

There is a relation between limited values of the average mixed moments of this sequence and values of moments of the limited random variable n, assuming that it exists and is absolutely continuous.

Theorem 2. The first r moments vk(S) = lim vk(S(n)), k = 1, 2,..., of a random variable n

are limited then and only then when the first r average mixed moments mk(n), k = 1, 2,... ,r of the sequence S are limited and vm(S) = hm(mn).

Proof. The statement follows from the following relation:

/w /• w m m /• w

hm(x)^v(x) dx = / aixl^v(x) dx ai xl^v(x) dx = hm(mn).

-w J—w i_0 i_0 —w

2. Rademacher random variables

The set Si of sequences of random variables £ = (Çt)teN, where £t(w) G X(9 = 1) = X(1), w G Qt with P(£t = 9) = pt and P(£t = —9) = 1 — pt = qt is so defined that for any sequence £ from this set the following conditions are fulfilled:

1n

1. - E ESt ^ 0,

n t=1 n^w

2. there is a weak limit S1/2(S) having a non-degenerate distribution

1 "

t=1

S1/2{C(n)) = ^E^ SI»®' v 1

Let us also introduce E4 C S1. It is a set of sequences of random variables with average links (shortly sal ) S = (St); St(w) € X(1), w € Qt, ESt = 0. They are built according to the Theorem 3.1 [7] for sequences S € E4 with a = 1/2. Sequences S have the following properties:

i. Distribution functions of random variables S1/2(S) and S1/2(£) coincide.

ii. vm(S) = vm(S) and Vm(S) = ;Vm(S), ym € N.

Theorem 3. Let a sequence S € E4 be defined. Then n = S1/2(S) is an absolutely continuous random variable with the density distribution function

^ 00

1 sr^

Vn (x) = —=e 2 }. Vm(S) ■ hm(x), yx € R. (9)

V 2n —'

m=0

Proof. Let us consider sequence S € ^ and sequence S € E4 associated with it. Let us consider random variables

1 n n

S1/2 {^(n)) = = J2 nt = S(n(n)),

n

t=1 t=1

where S(n) is the finite subsequence of sequence S, nt = 0nSt, 0n = —=.

n

2k n

Random variable S1/2(&n)) can take the values xn = —; k = 0,1,.. .,n and, in addition,

n

ES1/2(S(n)) = 0.

The probability that this random variable falls in the interval (-x>, b) is

P(S1/2(S(n)) <b)=J2 P(S1/2(S(n)) = xn)=J2 P(S(fi(n)) = xn),

xn<b xn<b

Here b is some real number. Using relations (1) and (3) we obtain

C k n

P(S1/2(i(n)) = xn) = P(S(n(n)) = xn)=2nJ2 0—mv m (n(n)) ■ Bn(m, k)

and

ps /2(S(n)) = x.n ) =

n 2n

2n

m=0

C k n ^

P{S1/2{£(n)) = xn )= 2n v'm{i(n)) • Bn{m,k).

m=0

Then we have

Ck n Ck n P{Sl/2{i(n)) <b)= Y. #E *m{£(n))Bn,m{k) = E #E vm{i(n))^n,m{k), (10)

2k-n <b m=0 2 k-n <b m=0

■/n

where yn,m(k) are the Kravchuk orthogonal polynomials (see [8,9]). They satisfy the following relations:

Bn(m k) 1

lim pn,m(k) = \im = lim yn,m(xn) = Hm(x) = hm(x),

where xn ^ x, x G R. Because £ is strictly stationary sequence with E£t = 0 the strong law of large numbers is applicable to it (see, for example, [10, p. 438]). Then for any e > 0 we have

P(sup IS^im))! > e) = N sup ^ ^it > ej m^n \m^n m t=i J

-> 0.

Moreover

P sup

\ m^n

1 S1(£(m))

) J k 1 \

> e = P sup--- > e

J \m^n m 2 J

r\ t r k almost sure 1

^ e -> 0, where from--> — = p,

>œ n n^œ 2

k

where random variable — is the frequency of positive elements in the sequence £(n). n

Let us consider the expression

C k 1

= Cnpkqn-k, where p = q =2.

Asymptotically equivalent conversions of this expression as well as the sums of the form

J2 Ckpkqn-k, where p = q =2 k2

k p

and--> p are considered in the proof of de Moivre-Laplace limit theorem (see, for example,

n n^tt

[10,11]). These conversions are also applicable in our case.

2

As a result, putting Axn = and substituting k for xn in (10), we have

n

_ ^x n

P(n <b) - lim \ e-^ \ vm(£(

n) ) • yn,m(xn ).

xn<b m=0

Let us note that

lim vm(£(n)) = vm(£), m = 1, 2,... .

n^tt

The expression for P(n < b) can be rewritten in the following way:

( 1 n " \

P(n < b) - lim —= e 2 y] vm(£(n)) • yn,m(xn) )Axn.

xn<b \ v m=0 /

Due to the existence of the function Fn and and taking into account integrability of sums

V2ne ~ .

^ Axn _XL 1 fx -, w _ lim > e 2 = ^^ e 2 dy.vx G R,

I.-von . foZ . foZ

n^œ

we obtain that the sum

n

lim y] Vm(£(n))<fn,m(xn )

n

, v "m\<>(n) m=0

necessarily exists and it is finite. Then, using the notation x = lim xn and taking into account

n^w

that lim n m(xn) = hm(x), we get that the following inequalities

n^w '

nw

0 < lim y] vm(£(n))Pn,m(xn) = Y^ vm(£)hm(x) < X

n^w i—' i—'

m=0 m=0

are true for any real x. It also follows that Vm(S), m = 1, 2,... exist and they are finite. As a result, we have

/x 1 f x y2 w

Vn (y) dy = ^^ e— vm(S) ■ hm(y) dy

-w V2n J —w m=0

for any x € R. □

Corollary 1. Let us assume that sequence S € E4 is given. Then the random variable n = S1/2(S) has all moments and they are finite if its distribution density vn is a continuous function on the whole real axis.

Proof. It follows from the continuity of vn that the series

Vn(x) = —^ y vm{£)hm{x)e v —'

v m=0

converges uniformly on the whole real axis. It allows us to integrate this expression term by term and obtain

Lk{n)= hk{x)y,n{x) dx = y^vm{£) hk{x)hm{x)e 2 dx = Vk{£).

J-<X V m=0 J-<X

Jk (H) = I 'lk (x)Vn (x) dx = 2_^ Vm(

m=0

Taking into account theorem 2, this expression proves the statement. □

Corollary 2. Let us assume that sequence S € E4 is given. Then the random variable n = S1/2(S) has the standard normal distribution then and only then when

lim Vm(S(n)) = 0.

n^w

Proof. It follows from theorem 3 that the density of the random variable n for an arbitrary value of the argument x € R is expressed by (9):

i OO

—= e 2 > vm(S) ■ hm(x) = —= e 2 ^ vm(S) = vm(S )=0 with m > 1. V 2n —v, v 2n

m=0

This proves the statement. □

In conclusion let us consider a set S0 of random variables sequences S = (St.)t.eN, where St(w) € XB = {0; 1}, w € Qt. They are defined in the following way: for any sequence S from this set the following conditions are true:

1) ESt = 2, yt € N; i.e p = P(St = 1) = 2,q =1 - P = P(St = 0) = 1;

2) there is a weak limit S^2(S) of the sequence S^2(S(n)) that has a non-degenerate distribution n

S1/2(£(n)) = -—= E It ^ S 1/2(1), here St = St .

V n n^w ^pq

2

2

For any sequence £ G S0 we construct a sequence 7 = (it)t£N, 7t = 2£t - 1; Vt G N. It follows from the features of the sequence £ G S0 that sequence 7 satisfies the following conditions:

1n

1) EYt = 0, Vt G N, moreover — J2E^t ^ 0,

n t=1 n^tt

2) there is a weak limit S1/2(j) of the sequence S1/2(j(n)) that has a non-degenerate distribution

1n

S1/2(7(n)) = ^Y] 7t ^ S1/2(7) \ n '' n^tt ' v t=1

and

FS1/2(i)(x)= FS1/2(Y)(x), Vx G R. (11)

Indeed, because p = q = 2 we have Vx G R and

FS„MV„x = P( —TS ± £—£t < x) = P(—±™ - V < x) = P(—± Y. < x).

This proves relation (11).

Then sequence 7 is an element of the set E4 and it satisfies the conditions of theorem 3. This means that

00

VSi/2(Y) = e 2 V vm(7) • hm(x), Vx G R. V 2n —'

v m=0

The relationship between Vm(^) and cVm(£) is

e(2 * ^ - 2))(2 * (£t2-1))...(2 * (£t. - 2;

vi(7) = e(2 * (£t! - 2))(2 * (£t2 - 2)) ... (2 * (£tm - = 2mcv/(£)

and it means that Vm(7) = 2mcVm(£). As a result, we obtain the following statement.

Corollary 3. Let us assume that sequence £ G S0 is given. Then n = S1/2 (£) is an absolutely continuous random variable with the distribution density function given by

^ 00

In(x) = —= e-x V 2mcVm(£) • hm(x), Vx G R. (12)

m=0

3. Lattice random variables

Let us consider a sequence of lattice random variables £ = (£t)teN, £t ■ ^ {-Os, ■ ■ ■, O(2k-

s),..., Os} = X(O, s); k = 0,1, ■ ■ ■, s. We assume that P(£t = O(2k - s)) = pt(k); J2 Pt(k) = 1.

k=0

As before E2 is a set of sequences of random variables £ = (£t), £t ■ Qt ^ X(O, s), t = 1, 2, ■ ■ ■. Arbitrary sequence £ from this set satisfies the following conditions:

n

1) n £ E£t ^ 0, t=1

2) there exists a nondegenerate random variable n = S1/2(S) such that

1n

S1/2 {&(n)) = —n E &

^ n.

n

t=1

Let us denote the set S = (St); St ■ ^ X(0, s), ESt = 0 by S2 C S2. The existence and construction of such sequences is described in theorem 3.2^ [7]. The sequences have the following property:

FV2(i)(x) = F^«)^ yx € R. (13)

Let us consider limit distribution properties of a random variable n = S1/2(S), where S € S2. We use the fact that every lattice random variable St ■ ^ X(0, s) can be represented as a sum of s Rademacher random variables jT ■ Q'T ^ X(1):

ts

St = 0 J2 Yt . (14)

T = (t—1)s + 1

According to Theorem 3.21 [7] we can construct sequence S = (St.)t.eN,St ■ ^t ^ X(0,s) and ESt = 0 that satisfies (13). Every subsequence S(n) of the sequence S can be related to a subsequence of Rademacher random variables Y(ns) such that

ts

L = 0 J2 Yt yt € In.

T = (t—1)s+1

Let us consider the manner in which the distribution density of random variable n = S1/2(S) can be expressed in terms of mixed moments of the sequence y = (yt)teN.

Theorem 4. Let us assume that sequence S € S2 is given. Then n = S1/2(S) is an absolutely continuous random variable with the distribution density function

oo

1 2 / x \

Vn(x) = e—^ £ vm(Y) ■ Myx € R,

v m=0 v

where y = (Yt), Yt ■ ^ X(1) is the sequence of Rademacher random variables that satisfies (14).

Proof. Let us consider the sequence y = (yt)teN, ) that satisfies (14). For a fixed value s the following relations are true:

1 n 0 ns 1 ns 1 k

1V ESt-► 0 ^ — V Eyt ^ 0 ^ — V Eyt 0 ^ 1V Eyt-► 0.

n z—' n^w ns z—' n ns z—' n k z—' k^w

t=1 T=1 T=1 T=1

Because \EYk\ < 1 yk € N and, assuming that n' = msx{n\ns < k}, we obtain

1 k i n S k

1

k\J2eyt < ^\Eeyt + -1- E EYt < E

k \ L—' n S —' n s L—' n s —'

t =1 t =1 t=n's +1 t =1

1

+ - -► 0.

n' k^^

t There is a misprint in the article. The corrected theorem: Let the sequence § = (§t)teN, §t : Qt ^ X(d,s) be defined on (Qn, An, Pj). For some a € (0; 1) a random variable Sa(§) ) is defined and it is a weak limit of the sums Sn>a (§). Then there exists § = (§t)teN, §t : Qt ^ X(0, s), such that ....

In addition, because

1 n ns

-^J^it = e—s

, /71. < J , /71. Q < J

t=1

(15)

T =1

1n

and, taking into account that the weak limit for ^^ J2 £t exists, then there exists the weak

n t=1

11 n 1 k

_ ^ !T ^ £t. One can conclude that sums ^^ £ 7T weakly converge

^Jns T =1 0\Js \JSj t=1 Vk t =1

1 ns

limit for ,_ J2 Y-

to some random variable. Let us assume that n is the weak limit of sequence

1n

S1/2 (£(n)) = -^J2£t * t=1

and n'' = n' +1. Then from the definition of weak convergence we obtain that in every continuity point x G R of the distribution function of the limit random variable n the following relation is true:

1

Taking into account

Fn (x) = lim P

(—= Y1 yt <x ).

ns

-k §>=-—k (§^ - „V

and

-k 5YT=-k is >+r£yt

i

—k

we obtain that

P

n ''s 1 \fk y/n''

n's 1

T

Yt - J2 Yt

t =k+1

J2yt + E

t=k + 1

J2Yt - Yt ) <x) < P Yt <x

it=1 T=k+1

S ^ <x)

^ P

J2Yt + J2 Yt

<x

,T =1

t=k + 1

for x ^ 0.

It follows that

1 1

Yt <x +

P

n s

t =1

and for x < 0 we have

P

1 1

<x +

< P( ^ t YT <x) < P > P( ^ S YT <x) > P

:EYt <x -—=

1 1 ) y=^YT <x -—= ) .

Therefore we obtain weak convergence when k ^ œ. Thus all the conditions of Theorem 3 are fulfilled for the sequence y = (jT)TeN ) and we can use it to obtain the density distribution function of the limit probability distribution. Let ni be the weak limit of the sequence

1n

S1/2 (Y(n)) = —^J2Yt.

T

Y

T

T

1

1

1

T

T

T

T

Then, taking into account (15), we obtain that the value of random variable n is related to the value of random variable n1 by the equation

n = 0^fs ■ n1. Then from Theorem 3 we obtain that yx € R

1 x2 w

Vni (x)= —,e ^ E vm(Y) ■ hm(x). v 2n —'

m=0

Finally we obtain that

1 2 w

Vn(x) = Ve^sni(x) = —n0rse— ^ E vm(Y) ■ M9—s) .

□

One should note that when 0 = — the density distribution function of random variable n

Vs

has the simplest form. In this case

^ oo

1 V—v

Vn (x) = ^= e 2 7. vm (Y) ■ hm (x) V 2n —'

m=0

for any x € R.

Let us consider how to change from mixed moments of the sequence Y to mixed moments of the sequence S . First of all, let us note that mixed moments of sequences Y and Y are the same (it follows from Theorem 3.1 [7] ).

For convenience we identify the elements of the sequence Y with the help of two indexes. The first index t defines the number of an element in sequence S . The second index defines the number of an element in the sequence Y in representation of the element St so that

s—1

St = 0j2Ytj.

j=0

Then, taking into account results obtained above, we have

vm (S(n)) _ vIm (S(n))

but

and

Hence

Vm (&(n))

vIm (i(n)) = E n & = emE n (E Ytj ) = emsmEY1,0Y2,0 ■ ■ ■ Ym,0, teim teim \j=1

emSmEY1,0Y2,0 . ..Ym,0 = emsmvim {Y(ns)) = emSmVm{Y(ns)) = emsmvm{Y(ns)).

0msmvm(Y(ns)) ¡C'

vm(Sin)) = -jCnl{ns)) = 0msmvm(Y(ns))^ Cm .

Taking into account (8), in the limit n ^ x we obtain

vm{Y) = ems ^ vm{Y) (16)

which leads to

Then the density of the random variable n G R is

-e 2e2s x

"ni(x) = e-2f2s m=0ml• hm{

Corollary 4.1. Let us assume that sequence £ £ E2 is given. Then random variable n

1 n

£t ^ n

\/n z—' n^x v t=1

has the normal distribution with parameters En = 0 and Dn = 92s then and only then when

Vm(£) = lim vm (£(n)) =0, m > 1.

Proof. It is similar to the proof of Corollary 2. □

Let us consider two special cases: the expression for the density of the sum of sequence £ £ E2

when 9 = — and 9 = —=. In the first case the change scale of x is conserved but the values V s s^s

of mixed moments are changed. In the second case the values of the moments are conserved but the change scale of x is changed.

Corollary 4.2. Let us assume that sequence £ £ E2 is given and 9 = —=. Then the random

Vs

variable n

1 n

&t ^ n

v t=1

for any x £ R has the density distribution function

i \ 1 - ^ i)m(i) , , S

Vv (x) = e 2 -^ • hm(x).

V2n

=0

Corollary 4.3. Let us assume that sequence £ £ E2 is given and 9 = ——. Then the random

S\/S

variable n

1n

^Y, &t ^ n

\/n z—'

v t=1

for any x £ R has the density distribution function

, x

s (sx)2 v *

"n(x) = —== e 2 V vm(£) • hm(sx).

v m=0

References

[1] Y.V.Nesterenko, Y.U.Nikitin, Scientific conference commemorating academician Y.V.Lin-nik, Vestnik of the St. Petersburg University: Mathematics, 1(2005), no. 4, 3-6 (in Russian).

[2] I.A.Ibragimov, Y.V.Linnik, Independent and stationary sequences of random variables, Moscow, Science, 1965 (in Russian).

[3] A.G.Grin', Norming Sequences in the Limit Theorems for Weakly Dependent Variables, Theory Probab. Appl., 36(2), 272-288.

[4] A.G.Grin', On Strong Attraction of Stationary Sequences to a Normal Law, Theory Probab. Appl., 44, no. 4, 768-775.

[5] A.G.Grin', On the Minimal Condition of Weak Dependency in the Central Limit Theorem for Stationary Sequences, Theory Probab. Appl., 47(3), 506-510.

[6] A.G.Grin', On minimal conditions of the weak dependence in limit theorems for stationary sequences, Theory Probab. Appl., 54(2010), no. 2, 307-317.

[7] S.V.Chebotarev, About sequences of random variables with averaged links, Vestnik AltGPA, seriya: estestvenye i tochnye nauki, 7(2011), 28-37 (in Russian).

[8] G.Szego, Orthogonal polynomials, Moscow, Fizmatgiz, 1962 (in Russian).

[9] S.V.Chebotarev, About the features of Kravchuk polynomials, Vestnik Barnaulsgogo Gos. Ped. Univ., seriya: estestvenye i tochnye nauki, 2(2002), 53-58 (in Russian).

[10] A.N.Shiryaev, Probability, Moscow, Nauka, 1989 (in Russian).

[11] A.A.Borovkov, Probability Theory, Moscow, Nauka, 1976 (in Russian).

О предельном распределении сумм случайных величин

Сергей В. Чеботарев

Рассмотрены центрированные последовательности радемахеровских и решетчатых случайных

1 п

величин, имеющие нетривиальный слабый предел сумм > Для них найден общий вид

Vй

у г=1

предельного распределения. Показано, что вид предельного распределения зависит лишь от усредненных смешанных моментов первого порядка, характеризующих случайные величины последовательности, причем в случае решетчатых случайных величин имеется в виду последовательность радемахеровских случайных величин, в которую можно 'разложить элементы рассматриваемой последовательности.

Ключевые слова: последовательности случайных величин, сумма случайных величин, сумма зависимых случайных величин, предельное распределение сумм случайных величин.