An almost sure limit theorem for α-mixing random fields

(2009) pp. 123–132

http://ami.ektf.hu

An almost sure limit theorem for α -mixing random fields

Tibor Tómács

Department of Applied Mathematics Eszterházy Károly College, Eger, Hungary

Submitted 15 September 2009; Accepted 3 November 2009

Abstract

An almost sure limit theorem with logarithmic averages forα-mixing ran- dom fields is presented.

Keywords: Almost sure limit theorem, multiindex, random field, α-mixing random field, strong law of large numbers

MSC:60F15, 60F17

1. Introduction

LetNbe the set of the positive integers,Rthe set of real numbers and Bthe σ-algebra of Borel sets ofR. Letδxbe the unit mass at pointx, that isδx: B →R, δx(B) = 1 ifx∈B andδx(B) = 0 ifx6∈B. Denote−→^w µthe weak convergence to the probability measure µ. In the following all random variables defined on a fixed probability space(Ω,F,P). Almost sure (a.s.) limit theorems state that

1 Dn

n

X

k=1

dkδζk(ω)

−→w µ as n→ ∞, for almost every ω∈Ω,

where ζk (k ∈N) are random variables. The simplest form of it is the so-called classical a.s. central limit theorem, in which ζk = (X1 +· · ·+Xk)/√

k, where X1, X2, . . . are independent identically distributed (i.i.d.) random variables with expectation 0 and variance 1, moreoverdk = 1/k,Dn= lognandµis the standard normal distributionN(0,1). (See Berkes [1] for an overview.)

LetN^d be the positive integerd-dimensional lattice points, where dis a fixed positive integer. In this paper k= (k1, . . . , kd),n= (n1, . . . , nd), . . . ∈N^d. Rela- tions6, ≮, min,max, →etc. are defined coordinatewise, i.e. n→ ∞ means that

123

ni→ ∞for alli∈ {1, . . . , d}. Let|n|=Qd

i=1niand|logn|=Qd

i=1log+ni, where log+x= logxifx>eandlog+x= 1ifx < e. The general form of the multiindex version of the a.s. limit theorems is

1 Dⁿ

X

k6n

d^kδζ_k(ω)

−→w µ as n→ ∞, for almost every ω∈Ω,

where {ζk,k∈ N^d} is a random field (multiindex sequence of random variables).

In the multiindex version of the classical a.s. central limit theorem Xⁱ,i ∈ N^d i.i.d. random variables with expectation 0 and variance 1, ζ^k = P

i6kXⁱ/p

|k|, d^k = 1/|k|, Dⁿ= 1/|logn| and µ=N(0,1). It is well-known that generally the multiindex cases are not direct consequences of the corresponding theorems for ordinary sequences.

Fazekas and Rychlik proved in [5] a general a.s. limit theorem for multiindex sequences of metric space valued random elements. Tómács proved in [8] an a.s.

central limit theorem form-dependent random fields. In this paper we shall prove an a.s. limit theorem with logarithmic averages for α-mixing random fields (The- orem 2.5). Its onedimension version for µ = N(0,1) is proved by Fazekas and Rychlik (see [4, Proposition 3.2]). In the proof of Theorem 2.5 we shall use a mul- tiindex strong law of large numbers (Theorem 2.1). In the proof of Theorem 2.3 we shall follow ideas of Berkes and Csáki [2].

Throughout the paper we use the following notation. LetR+be the set of the positive real numbers. Ifa1, a2, . . .∈Rthen in case A=∅ letmaxk∈Aak = 0and P

k∈Aak= 0. Let[A]be the closure ofA⊂Rand∂A= [A]∩[A].

If ξ is a random variable, then let µξ denote the distribution of ξ, kξk^∞ = inf{c∈R: P(|ξ|6c) = 1}and σ(ξ) ={ξ⁻¹(B) :B ∈ B}.

In the following let{c⁽ⁱ⁾_k ∈R+, k∈N}be increasing sequences withc⁽ⁱ⁾_k+1/c⁽ⁱ⁾_k = O(1),limn→∞c⁽ⁱ⁾n =∞for eachi= 1, . . . , d, and the sequences{d⁽ⁱ⁾_k ∈R+, k∈N} have the next properties: d⁽ⁱ⁾_k 6 log(c⁽ⁱ⁾_k+1/c⁽ⁱ⁾_k ) for all k ∈ N and i = 1, . . . , d, moreoverP∞

k=1d⁽ⁱ⁾_k =∞for eachi= 1, . . . , d. Letd^k=Qd

i=1d⁽ⁱ⁾_ki,Dⁿ=P

k6nd^k andD⁽ⁱ⁾ni =Pni

k=1d⁽ⁱ⁾_k .

2. Results

Theorem 2.1. Let{ξⁱ,i∈N^d} be a uniformly bounded random field, namely there exists c ∈ R+ such that |ξⁱ| 6 c a.s. for all i ∈ N^d. Assume that there exist c1, c2, ε∈R+ andα^k,l∈R(k,l∈N^d)such that

X

l6n

X

k6n

dkdlαk,l6c1Dn² d

Y

i=1

logD⁽ⁱ⁾_ni−1−ε

(2.1)

for all enough largeni ∈N, and

|Eξ^kξ^l|6c2





d

Y

i=1

log+log+

c⁽ⁱ⁾mi

c⁽ⁱ⁾_hi

!−1−ε

+α^k,l



 (2.2)

for each k,l∈N^d, whereh= min{k,l}and m= max{k,l}. Then

1 Dⁿ

X

k6n

d^kξ^k →0 as n→ ∞ a.s.

Definition 2.2. Theα-mixing coefficient of the random variablesξandη is α(ξ, η) =α σ(ξ), σ(η)

= sup

A∈σ(ξ) B∈σ(η)

|P(AB)−P(A) P(B)|.

Theorem 2.3. Let {ζ^k,k∈N^d} be a random field. Assume that there exist ran- dom variables ζ^h,l (h6l)andc1, c2, c3, ε∈R+ such that

|ζ^k−ζ^h,k|>c1 a.s. ∀h,k∈N^d for which h6k, (2.3)

E min

(ζ^l−ζ^h,l)²,1 6c2 d

Y

i=1

log+log+

c⁽ⁱ⁾_li c⁽ⁱ⁾_hi

!^−2−2ε

(2.4) for allh,l∈N^d for which h6l, and

X

l6n

X

k6n

d^kd^lα^k,l6c3Dn² d

Y

i=1

logD⁽ⁱ⁾_ni−1−ε

(2.5)

for all enough large ni ∈N, where α^k,l=α(ζ^k, ζ^t,l) with t= min{k,l}. Then for any probability distribution µthe following two statements are equivalent:

(1) 1 Dⁿ

X

k6n

d^kδζ_k(ω)

−→w µasn→ ∞, for almost every ω∈Ω;

(2) 1 Dn

X

k6n

d^kµζ_k

−→w µasn→ ∞.

Definition 2.4. Theα-mixing coefficient of the random field{Xⁿ,n∈N^d}is

α(k) = sup

n

α



 [

i6n

σ(Xⁱ), [

i≮n+k

σ(Xⁱ)



, k∈N^d.

Theorem 2.5. Let {Xⁿ,n∈N^d}be an α-mixing random field with mixing coeffi- cient

α(k)6 c

|logk| (2.6)

for all k ∈ N^d, where c ∈ R+ is fixed. Let Sⁿ =P

k6nX^k and σn² = ES²n >0.

Assume that EXⁱ= 0and EXi²<∞ for alli∈N^d, moreover there exist c1, c2∈ R+ andβ >2/log 2such that

|Sl|>c1σk a.s. ∀l,k∈N^d for which l6k (2.7) and

E min Sr²

σl²

,1

6c2

|h|

|l| β

∀h,l∈N^d for which h6l, (2.8) where r = 2h if 2h < l and r = l otherwise. If µζ_n

−→w µ as n → ∞, where ζⁿ=Sⁿ/σⁿ andµis a probability distribution, then

1

|logn| X

k6n

1

|k|δζ_k(ω)

−→w µ as n→ ∞, for almost every ω∈Ω.

3. Lemmas

You can find the proof of the next lemma in [6].

Lemma 3.1 (Covariance inequality). If ξ and η are bounded random variables, then

|cov(ξ, η)|64α(ξ, η)kξk^∞kηk^∞.

The proof of the next lemma follows from that of Theorem 11.3.3 and Corol- lary 11.3.4 in [3].

Lemma 3.2. Let BL denote the set of all bounded, real-valued Lipshitz function on R. If µ and µn are distributions (n ∈ N), then there exists a countable set M ⊂BL (depending onµ) such that the following are equivalent:

(1) µn

−→w µ asn→ ∞; (2) R

gdµn→R

gdµasn→ ∞ for allg∈M.

Lemma 3.3 (Theorem 1 of [7], p. 309). If µ and µn are distributions (n ∈ N), then the following are equivalent:

(1) µn

−→w µ asn→ ∞;

(2) µn(A)→µ(A)asn→ ∞ for allA∈ B for which µ(∂A) = 0.

Lemma 3.4. If µ and µⁿ are distributions (n ∈ N^d) and µⁿ −→^w µ as n → ∞,

then 1

Dⁿ X

k6n

d^kµ^k−→^w µ as n→ ∞.

Proof. ByP∞

ki=1d⁽ⁱ⁾_ki =∞we have 1

Dⁿ X

m6k6n

d^k =

d

Y

i=1

P

mi6ki6nid⁽ⁱ⁾_ki P

ki6nid⁽ⁱ⁾_ki →1 as n→ ∞ ∀m∈N^d, which implies, that

1 Dⁿ

X

k6n km

d^k= 1− 1 Dⁿ

X

m6k6n

d^k→0 as n→ ∞ ∀m∈N^d. (3.1)

Letf:R→Rbe a bounded and continuous function andK= sup_x∈R|f(x)|. Then

Z

fdµⁿ− Z

fdµ

6 Z

Kdµⁿ+ Z

Kdµ= 2K, (3.2)

moreover byµⁿ−→^w µand (3.1), for anyε >0there exists n(ε)∈N^d such that

Z

fdµn− Z

fdµ

< ε

2 (3.3)

and 1

Dn

X

k6n kn(ε)

d^k< ε

4K (3.4)

for all n>n(ε). With notationγⁿ= _D¹

n

P

k6nd^kµ^k the inequalities (3.2), (3.3) and (3.4) imply, that

Z

fdγⁿ− Z

fdµ

6 1 Dⁿ

X

k6n

d^k Z

fdµ^k− Z

fdµ

= 1 Dⁿ

X

k6n kn(ε)

d^k Z

fdµ^k− Z

fdµ

+ 1 Dⁿ

X

n(ε)6k6n

d^k Z

fdµ^k− Z

fdµ

< 1 Dⁿ

X

k6n kn(ε)

d^k·2K+ 1 Dⁿ

X

n(ε)6k6n

d^k·ε 2 <ε

2 +ε 2 =ε

for alln>n(ε). This fact implies the statement.

4. Proof of the theorems

Proof of Theorem 2.1. By (2.2) and (2.1) we have

E



 X

k6n

d^kξ^k





2

6X

k6n

X

l6n

d^kd^l|Eξ^kξ^l|

6c2

X

k6n

X

l6n d

Y

i=1

d⁽ⁱ⁾_kid⁽ⁱ⁾_li log+log+

c⁽ⁱ⁾mi

c⁽ⁱ⁾_hi

!−1−ε

+c2

X

k6n

X

l6n

d^kd^lα^k,l

62c2 d

Y

i=1

X

ki6li6ni

d⁽ⁱ⁾_kid⁽ⁱ⁾_li log+log+

c⁽ⁱ⁾_li c⁽ⁱ⁾_ki

!^−1−ε

+c2c1Dn² d

Y

i=1

logD⁽ⁱ⁾_ni−1−ε

(4.1)

for all enough largeni. Now assume that(ki, li)∈A⁽ⁱ⁾ni, where A⁽ⁱ⁾ni =n

(ki, li) :ki 6li6ni andc⁽ⁱ⁾_li /c⁽ⁱ⁾_ki >exp ^qD(i)_nio . Thenlog+log+

c⁽ⁱ⁾_li /c⁽ⁱ⁾_ki

> ¹₂logDn⁽ⁱ⁾i, which implies, that

X

(ki,li)∈A⁽ⁱ⁾_ni

d⁽ⁱ⁾_kid⁽ⁱ⁾_li log+log+

c⁽ⁱ⁾_li c⁽ⁱ⁾_ki

!^−1−ε

62^1+ε

logD_n⁽ⁱ⁾_i−1−ε X

(ki,li)∈A⁽ⁱ⁾_ni

d⁽ⁱ⁾_kid⁽ⁱ⁾_li 62^1+ε

D⁽ⁱ⁾_ni2

logD⁽ⁱ⁾_ni−1−ε

. (4.2)

If(ki, li)∈Bn⁽ⁱ⁾i, where B_n⁽ⁱ⁾_i =n

(ki, li) :ki6li6ni andc⁽ⁱ⁾_li /c⁽ⁱ⁾_ki <exp ^qD_ni⁽ⁱ⁾o ,

then with notationMi= sup_k(c⁽ⁱ⁾_k+1/c⁽ⁱ⁾_k ), we get logc⁽ⁱ⁾_li+1

c⁽ⁱ⁾_ki = logc⁽ⁱ⁾_li+1

c⁽ⁱ⁾_li + logc⁽ⁱ⁾_li

c⁽ⁱ⁾_ki <logMi+ q

Dn⁽ⁱ⁾i.

Thus we have the following inequality, whereB_n⁽ⁱ⁾_i,ki =n

li: (ki, li)∈Bn⁽ⁱ⁾i

o.

X

(ki,li)∈B⁽ⁱ⁾_ni

d⁽ⁱ⁾_kid⁽ⁱ⁾_li log+log+

c⁽ⁱ⁾_li c⁽ⁱ⁾_ki

!^−1−ε

6 X

(ki,li)∈B_ni⁽ⁱ⁾

d⁽ⁱ⁾_kid⁽ⁱ⁾_li

6 X

(ki,li)∈B_ni⁽ⁱ⁾

d⁽ⁱ⁾_ki logc⁽ⁱ⁾_li+1 c⁽ⁱ⁾_li =

ni

X

ki=1

X

li∈B⁽ⁱ⁾

ni,ki

d⁽ⁱ⁾_ki logc⁽ⁱ⁾_li+1 c⁽ⁱ⁾_li

6

ni

X

ki=1

d⁽ⁱ⁾_ki

maxB⁽ⁱ⁾

ni,ki

X

li=ki

logc⁽ⁱ⁾_li+1 c⁽ⁱ⁾_li

=

ni

X

ki=1

d⁽ⁱ⁾_ki log

maxB⁽ⁱ⁾

ni,ki

Y

li=ki

c⁽ⁱ⁾_li+1 c⁽ⁱ⁾_li =

ni

X

ki=1

d⁽ⁱ⁾_ki log c⁽ⁱ⁾

maxB⁽ⁱ⁾

ni,ki

c⁽ⁱ⁾_ki

<

ni

X

ki=1

d⁽ⁱ⁾_ki

logMi+ q

Dn⁽ⁱ⁾i

6

ni

X

ki=1

d⁽ⁱ⁾_ki2 q

Dn⁽ⁱ⁾i = 2

D⁽ⁱ⁾_ni3/2

for all enough largeni. It follows from this inequality and (4.2) that

X

ki6li6ni

d⁽ⁱ⁾_kid⁽ⁱ⁾_li log+log+

c⁽ⁱ⁾_li c⁽ⁱ⁾_ki

!^−1−ε

62^1+ε

D⁽ⁱ⁾_ni2

logD⁽ⁱ⁾_ni−1−ε

+ 2

D⁽ⁱ⁾_ni3/2

62^1+ε

D⁽ⁱ⁾_ni2

logD⁽ⁱ⁾_ni−1−ε

+

D⁽ⁱ⁾_ni−1/2

62^2+ε

D⁽ⁱ⁾_ni2

logD⁽ⁱ⁾_ni−1−ε

(4.3) for all enough large ni. In the last step we use the inequality (D⁽ⁱ⁾ni)^−1/2 6 (logDn⁽ⁱ⁾i)^−1−ε, which follows from (D⁽ⁱ⁾ni)^1/2/(logD⁽ⁱ⁾ni)^1+ε → ∞ as ni → ∞. By (4.1) and (4.3) we get

E



 X

k6n

d^kξ^k





2

6const.

d

Y

i=1

D⁽ⁱ⁾_ni2

logD⁽ⁱ⁾_ni−1−ε

(4.4)

for all enough largeni. Let

ni(t) = minn

ni:D⁽ⁱ⁾_ni 6exp t^1+ε/21+ε o

and n(t) = n1(t1), . . . , nd(td)

. Sinceni(ti)→ ∞as ti → ∞, thus by (4.4) there existsT∈N^d, such that

EX

t>T



 1 Dn(t)

X

k6n(t)

d^kξ^k





2

6X

t>T

1 D²n(t)

const.

d

Y

i=1

D_n⁽ⁱ⁾_i(ti)2

logD⁽ⁱ⁾_ni(ti)−1−ε

6X

t>T

1 D²n(t)

const.

d

Y

i=1

D_n⁽ⁱ⁾_i(ti)2

t^−1−ε/2_i =const.

d

Y

i=1

∞

X

ti=Ti

t^−1−ε/2_i <∞,

which implies

1 Dn(t)

X

k6n(t)

d^kξ^k→0 as t→ ∞ a.s. (4.5)

For all n ∈ N^d there exists t ∈ N^d such that n(t) 6 n 6 n(t+1), where 1 = (1, . . . ,1)∈N^d. Thus the uniformly bounding implies

1 Dⁿ

X

k6n

d^kξ^k

6

1 Dn(t)

X

k6n(t)

d^kξ^k

+ 1 Dⁿ

X

k6n kn(t)

d^k|ξ^k|

6

1 Dⁿ(t)

X

k6n(t)

d^kξ^k + 1

Dⁿ X

k6n kn(t)

d^k·c

6

1 Dn(t)

X

k6n(t)

d^kξ^k

+c

1− Dn(t)

Dn(t+1)

a.s. (4.6)

The reader can easy verify that Dⁿ(t)/Dⁿ(t+1) → 1 as t → ∞, so by (4.5) and

(4.6) imply the statement of Theorem 2.1.

Proof of Theorem 2.3. Letg ∈ M, where M is defined in Lemma 3.2. Then there exists K>1 such that

|g(x)|6K and |g(x)−g(y)|6K|x−y| ∀x, y∈R. (4.7) We shall prove, that with notation ξ^k =g(ζ^k)−Eg(ζ^k) the conditions of Theo- rem 2.1 hold true. By (2.5) we get (2.1), moreover by (4.7) we have

|ξ^k|6|g(ζ^k)|+ E|g(ζ^k)|62K,

thus{ξ^k,k∈N^d} is a uniformly bounded random field. Now we turn to (2.2). Let t= min{k,l}. Lemma 3.1 and (4.7) imply

Eξ^k g(ζ^t,l)−Eg(ζ^l) =

cov g(ζ^l), g(ζ^t,l)

64K²α^k,l. (4.8) On the other hand with notationη^k,l=g(ζ^l)−g(ζ^t,l)

|Eξ^kη^k,l|=

cov g(ζ^k), η^k,l

6 Eg²(ζ^k) Eη²k,l

1/2

. (4.9)

It is easy to see that g(x)−g(y)2

64K²min

(x−y)²,1 , thus Eη²k,l64K²min

(ζ^l−ζ^t,l)²,1 . (4.10) By (4.7) and (2.3) we haveg²(ζ^k)6K²(1 + 1/c1)² and

g²(ζ^k)< K²(c1+ 1)²=K²

1 + 1 c1

2

·c²₁6K²

1 + 1 c1

2

(ζ^k−ζ^t,k)², which implyg²(ζ^k)6const.min

(ζ^k−ζ^t,k)²,1 a.s. Using this inequality, (4.10), (4.9) and (2.4) we get the following.

|Eξkηk,l|6const. E min

(ζk−ζt,k)²,1 E min

(ζl−ζt,l)²,1 ^1/2 6const.

d

Y

i=1

log+log+

c⁽ⁱ⁾_ki

c⁽ⁱ⁾_ti ·log+log+

c⁽ⁱ⁾_li c⁽ⁱ⁾_ti

!^−1−ε

=const.

d

Y

i=1

log+log+

c⁽ⁱ⁾mi

c⁽ⁱ⁾_ti

!^−1−ε

, (4.11)

where m= max{k,l}. Since |Eξ^kξ^l|6|Eξ^kη^k,l|+

Eξ^k g(ζ^t,l)−Eg(ζ^l) , using (4.11) and (4.8) we have (2.2). Now applying Theorem 2.1 we get

1 Dⁿ

X

k6n

d^kξ^k→0 as n→ ∞ a.s. (4.12)

Letµn=_D¹

n

P

k6ndkµζ_k andµn,ω =_D¹

n

P

k6ndkδζ_k(ω) (ω∈Ω).

First assume that (2) is true, that isµⁿ −→^w µ as n→ ∞. Then Lemma 3.2 implies

Z

gdµⁿ→ Z

gdµ as n→ ∞, (4.13)

and (4.12) implies Z

gdµⁿ,ω− Z

gdµⁿ= 1 Dⁿ

X

k6n

d^kξ^k(ω)→0 (4.14)

asn→ ∞, for almost everyω∈Ω. By (4.13) and (4.14) we getR

gdµⁿ,ω→R gdµ as n→ ∞, for almost everyω∈Ω, thus by Lemma 3.2 we get (1).

Finally assume that (1) is true, that isµⁿ,ω

−→w µasn→ ∞, for almost every ω ∈ Ω. Let A ∈ B and µ(∂A) = 0. Then by Lemma 3.3 µⁿ,ω(A) → µ(A) as n→ ∞, for almost everyω∈Ω. It follows thatµⁿ(A) =R

µⁿ,ω(A) d P(ω)→µ(A) as n → ∞. Thus using Lemma 3.3 we get (2). This completes the proof of

Theorem 2.3.

Proof of Theorem 2.5. Letd⁽ⁱ⁾_k = 1/k,c⁽ⁱ⁾_k =k^{1/log 2},ε= (βlog 2−2)/2,ζ^k,l= ζ^l−S2k/σ^lif2k<landζ^k,l= 0ifk6land2k≮l. We shall prove that conditions of Theorem 2.3 hold. It is easy to see thatα^k,l6α(k)for allk,l∈N^d, whereα^k,l

is defined in Theorem 2.3. Therefore by (2.6) we have X

l6n

X

k6n

d^kd^lα^k,l6X

l6n

X

k6n

c

|k| · |l| · |logk|

=c

d

Y

i=1 ni

X

k=1

1 klog+k

! _ni X

l=1

1 l

!

. (4.15)

It is well-known thatPn k=1 1

k ∼lognandPn

k=1 1

klog+k ∼log logn, wherean ∼bn

ifflimn→∞an/bn= 1. So by (4.15) we have X

l6n

X

k6n

d^kd^lα^k,l6const.

d

Y

i=1

log logni·logni6const.

d

Y

i=1

(logni)²(log logni)^−1−ε

6const.

d

Y

i=1

(logni)²(logD⁽ⁱ⁾_ni)^−1−ε6const.D²n d

Y

i=1

(logD⁽ⁱ⁾_ni)^−1−ε

for all enough largeni, which implies (2.5). Using (2.8)

E min

(ζl−ζh,l)²,1 = E min

Sr²/σ²l,1 6const.

d

Y

i=1

log+log+

c⁽ⁱ⁾_li c⁽ⁱ⁾_hi

!^−2−2ε

for all h,l∈N^d for which h6l, wherer = 2hif2h<land r= lif h6l and 2h6<l, so we get (2.4). The reader can readily verify that (2.3) is hold as well.

Now applying Lemma 3.4 and Theorem 2.3, we have 1

P

k6n 1

|k|

X

k6n

1

|k|δζ_k(ω)

−→w µ as n→ ∞, for almost every ω∈Ω.

SinceP

k6n 1

|k|∼ |logn|, we get the statement.

References

[1] Berkes, I., Results and problems related to the pointwise central limit theorem, In: Szyszkowicz, B. (Ed.) Asymptotic results in probability and statistics, Elsevier, Amsterdam,(1998), 59–96.

[2] Berkes, I., Csáki, E., A universal result in almost sure central limit theory,Stoch.

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[3] Dudley, R.M., Real Analysis and Probability,Cambridge University Press, (2002).

[4] Fazekas, I., Rychlik, Z., Almost sure functional limit theorems,Annales Universi- tatis Mariae Curie-Skłodowska Lublin, Vol. LVI, 1, Sectio A, (2002) 1–18.

[5] Fazekas, I., Rychlik, Z., Almost sure central limit theorems for random fields, Math. Nachr., 259, (2003), 12–18.

[6] Lin, Z., Lu, C., Limit theory for mixing dependent random variables,Science Press, New York–Beijing and Kluwer, Dordrecht–Boston–London (1996).

[7] Shiryayev, A.N., Probability,Springer-Verlag New York Inc.(1984).

[8] Tómács, T., Almost sure central limit theorems form-dependent random fields,Acta Acad. Paed. Agriensis, Sectio Mathematicae, 29 (2002) 89–94.

Tibor Tómács

Department of Applied Mathematics Eszterházy Károly College

P.O. Box 43 H-3301 Eger Hungary

e-mail: tomacs@ektf.hu