Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 656257, 9 pages http://dx.doi.org/10.1155/2013/656257

Research Article

The Almost Sure Local Central Limit Theorem for the Negatively Associated Sequences

Yuanying Jiang1,2 and Qunying Wu1

1 College of Science, Guilin University of Technology, Guilin 541004, China

2 School of Statistics, Renmin University of China, Beijing 100872, China

Correspondence should be addressed to Yuanying Jiang; jyy@ruc.edu.cn Received 13 May 2013; Accepted 18 June 2013 Academic Editor: Ying Hu

Copyright © 2013 Y. Jiang and Q. Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, the almost sure central limit theorem is established for sequences of negatively associated random variables: (1/logn)J^c=1(I(ak < Sk < bk)/k)P(ak < Sk <hk) = 1, almost surely. This is the local almost sure central limit theorem for negatively associated sequences similar to results by Csaki et al. (1993). The results extend those on almost sure local central limit theorems from the i.i.d. case to the stationary negatively associated sequences.

1. Introduction

Definition 1. Random variables X1,X2,...,Xn, n > 2 are said to be negatively associated (NA) if for every pair of disjoint subsets A1 and A 2 of {1,2,..., n},

Cov(f! (X,, ieAj, f2 (Xj, jeA2))<0,

where f1 and f2 are increasing for every variable (or decreasing for every variable) such that this covariance exists. A sequence of random variables {Xt, i > 1} is said to be NA if every finite subfamily of {Xt, i > 1} is NA.

Obviously, if {Xj,i > 1} is a sequence of NA random variables and {ft,i > 1} is a sequence of nondecreasing (or nonincreasing) functions, then {ft(Xt),i > 1} is also a sequence of NA random variables.

This definition was introduced by the Joag-Dev and Proschan [1]. Statistical test depends greatly on sampling, and the random sampling without replacement from a finite population is NA, but it is not independent. NA sampling has wide applications such as those in multivariate statistical analysis and reliability theory. Because of the wide applications of NA sampling, the notions of NA random variables have received more and more attention recently. We refer to Joag-Dev and Proschan [1] for fundamental properties, Shao

[2] for the moment inequalities, and Wu and Jiang [3] for Chover's law of the iterated logarithm.

Assume that {Xn, n > 1} is a strictly stationary sequence of NA random variables with EX1 = 0,0 < EXt < <x>. Define Sn = I"=1 xj,

on .= Var S^

o2 := VarX1 +2^Cov(X1,X;-). (3)

(1) Newman [4] and Matula [5] showed that NA stationary sequences satisfy the central limit theorem (CLT) under a2 > 0, that is,

P(^n < x ) - O (x)

= 0(1).

(2) Applying Matula [6] and Wu's [7] methods, we can easily show that NA sequences satisfy the almost sure central limit theorem (ASCLT), that is,

lim -T-I \Sk < xak1'2} = O (x) a.s. Vx e R, (5)

where O(x) is the standard normal distribution function and /{A} denotes the indicator of the event A.

The ASCLT was stimulated by Brosamler [8] and Schatte [9]. Both were concerned with the partial sum of independent and identically distributed (i.i.d.) random variables with more than the second moment. The ASCLT was extensively studied in the past two decades and an interesting direction of the study is to prove it for dependent variables. There are some results for weakly dependent variables such as a, p, 0-mixing and associated random variables. Among those results, we refer to Peligrad and Shao [10], Matula [11], and Wu [7].

More general version of ASCLT was proved by Csaki et al. [12]. The following theorem is due to them.

Theorem A. Let {Xn,n > 1} be a sequence of i.i.d. random variables with E\X1\ < to, let EX1 = 0. ak, bk satisfy

-TO<ak <0<bk <to, k=1,2,..., (6)

and assume that

1 k3/2P (ak < Sfc < bk)

= O (log n), as n —> to, (7)

and then

I{"k <Sk <bk} log nf kP(ak <Sk < bk)

This result may be called almost sure local central limit theorem, while (5) may be called almost sure global central limit theorem. Hurelbaatar [13] extended (8) to the case of p-mixing sequences and Weng et al. [14] derived an almost sure local central limit theorem for the product of partial sums of a sequence of i.i.d. positive random variables under some regular conditions. For more details, we refer to Berkes and Csaki [15] and Foldes [16].

Our concern in this paper is to give a common generalization of (8) to the case of NA sequences.

In the next section we present the exact results, postponing some technical lemmas and the proofs to Section 3.

2. Main Results

Assume in the following that {Xn,n > 1} is a strictly stationary sequence of NA random variables with EX1 = 0,0 < EX2 < to. We consider the limit behavior of the logarithmic average

_f i h <Sk < bk} ~ log nf kP(dk <Sk < bk)

with -to < ak < 0 < bk < to, where the terms in the sum above are defined to be 1 if their denominator happens to be 0.

More precisely let {an,n > 1} and {bn,n > 1} be two sequences of real numbers and put

Pk ■= P (ak <Sk <bk), I{dk <Sk <bk}

if Pk = 0, if Pk = 0.

So we need to investigate the limit behavior of

:==f i k=\ K

under certain conditions.

In our considerations, we will need the following Cox-Grimmett coefficient which describes the covariance structure of the sequence

u(n):= sup £ |Cov (Xj,Xk)\, n e N U {0}. ^

We remark that for a stationary sequence of NA random variables

u(n) = -2£ Cov(Xl,Xk), neN. (13)

By Lemma 8 of Newman [4], we have u(0) < m and limn^OTU(n) = 0.

In the following, Çn ~ qn denotes £,n/qn ^ 1,n ^ m. Çn = 0(qn) denotes that there exists a constant c > 0 such that Çn

< c^n for sufficiently large n. ^he symbols c, c^, c2,..., stand for generic positive constants which may differ from one place to another.

Theorem 2. Let {Xn, n > 1} be a strictly stationary sequence of NA random variables with EX1 = 0, £|X1|3 < m and let a2 >

0. ak, bk satisfy (6). Assume that

fu (n)

and for some p > 1,

l<k<n kPk Pk = 0

Then we have

(log k)

: O ((log n)2(log logn)ß). (15)

where ^n is defined by (11).

Remark 3. Let ak = -to and bk = xak1'2 6 limit theorem (4), we have pk = P(Sk/<jk1/2 < x) ^ obviously (15) satisfies; then (16) becomes (5), which is the almost sure global central limit theorem. Thus the almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem.

Remark 4. The condition (15) is satisfied with a wide range of pk; for example, if

Pk = 0 or Pk Zc

(log log kf (log kf3

holds, then the condition (15) is satisfied. In fact, letting 0 < S < 1,we have

£ (log kf3

1<k<n kPk

<c£ (log logk)-ß(logk)'

(log k)

<c(loglog n) ß(log n)8 £ ^^

1< k< n

/ 1/2-u ^ ^ i 1/2-a

-c1k < ük < -c2k , c3k1/2-a <bk < c4k1/2-a,

where 0 < a < 1/7. Assume that (14) hold, and then we have (16).

3. Proofs

The following lemmas play important roles in the proof of our theorems. The proofs are given in the Appendix.

Lemma 6. Assume that {%n, n >1} are random variables such that

Sk >0, Etk = 1, k=l,2,..

and furthermore there exists dk > 0 such that Dn = YH=1 T <x>, Dn/Dn-1 ^ 1, and

Var (£dktk )<cD2n(log Dn) , (21)

with some fi > 1 and positive constant c, and then 1 n

„limo =1 a.s. (22)

Remark 7. Let dk = 1/k in (21), and then Dn = ^=1 1/k log n. Thus, if

Var < c(logn)2(log logn)-

with some ß > 1 and positive constant c, then

1 n 1 lim -— y1$k = 1

logn k

The following Lemma 8 is obvious.

< c(log log n) ß(log n)2 = o ((log n) (log log n)

In the given theorem below, we strengthen the condition (6) on ak and bk. Meanwhile, as a compensation, we do not need to impose restricting condition (15) on pk.

Theorem 5. Let {Xn, n > 1}bea strictly stationary sequence of NA random variables with EX1 = 0, | < <x>, and a > 0, and let ak and bk satisfy

Lemma 8. Assume that the nonnegative random sequence {¡,n,n > 1} satisfies (24) and the sequence {rjn,n > 1} is such that, for any e > 0, there exists a k0 = kQ(e, co) for which

(1-e)$k <nk <(1 + z)h> k>kQ. (25)

Then we have also

logn k

1 £ 1 1 L-^k = 1

The following Lemma 9 is an easy corollary to the Corollary 2.2 in Matula [5] under strictly stationary condition, which studies the rate of convergence in the CLT under negative dependence. It was also studied in Pan [17]. Of course this is the Berry-Esseen inequality for the NA sequence.

Lemma 9. Let {Xj, j e N} be a strictly stationary sequence of NA random variables with EX 1 = 0, £|X1|3 < <x>,a2 > 0 satisfying (14). Then one has

<x)-O(x) = o(n1/5). (27)

Lemma 10. If the conditions of Lemma 9 hold, ak and bk satisfy (19). Then one has

c[k a <pk < c'2ka (k> k0), where a is as (19).

Lemma 11. If the conditions of Lemma 9 hold, ak and bk satisfy (19), and a is as (19). Assume that = l3a/2, and then the following asymptotic relations hold:

£ -¡j-P(\SP\> el) = 0(logn),

1<k<l<n KLPl k<l-l"

1<k<l<nklPl (l-k-l«)1/5 k<l-l"

= O (logn)

£ —

1<k<l<n klPl k<l- f

a,-K- e,

(l-k- la)

-O( $)

= O (log n),

1<k<l<n klPl k<l-l"

bl-ak + £l )-O( (l-k-l«)1/2 ) ( l1/2

= O (logn).

The main point in our proof is to verify the condition (23). We use global central limit theorem with remainders and the following elementary inequalities:

|® (x) - O (y)\ < c \x - y\, for every x, ye R (33)

with some constant c. Moreover, for each k > 0, there exists q = q(fc), such that

> q \x - y\, for every x, y e R, \x\ + \y\ < k.

Let {Xn, n > 1} be a strictly stationary sequence of NA random variables with a2 > 0; we can immediately get an ~ no2, that is,

qn < Var (Sn) = o2n < c2,.

for some constant q, c2 > 0 and sufficiently large n.

Proof of Theorem 2. First assume that

bk - ak < ck1/2, k = 1,2,...

with some constant c. Let 1 < k < I and £k = k1/2(logk)1/3.

If either pk = 0 or pi = 0, then obviously Cov(«k, <^i) = 0, and so we may assume that pkpi = 0; then, we have

Cov (ak, a-i 1

PiPk 1

(P(ak <Sk <bk,al <Si <k)-pipk)

<— (p(ai -bk <Si -Sk <bi - a^ pi pk

ak <Sk < bk) - PiPk)

<PP(P(*-bk <Si-Sk <bi-,

Xp(ak <Sk <bk)-PiPk)

= P(p (ai -bk <Si -Sk <bi - ak) - Pi) Pi

<-1(p (ai -bk - h <Si <bi -ak + £k) Pi

-Pi + P(|Sk|> Sk))

<—(p (ai -bk - £k <Si < ai) Pi

+ p(bi <Si <b-ak + Bk) +p(|Sk|> Bk)).

Applying Lemma 9, (33), (35), and (36) and noting that ek = k1/2(logk)1/3, we obtain

P (ai -bk - £k <Sl <ai) + P (bi <Si <bi - ak + £k)

f ^ f ^ f fy -bk - £k < (O ( — i k

+ )-0(h ))+c _L

< bk + £k + -ak + £k + c_L = bk -ak + + CJ_

< Ot Oi I1'5 Oi Oi I1'5

e^ M (k1/2(logk)^ J_

< C ( 11/2 + 11/2 + 11/5 ) < c ( 11/2 + 11/5 J ■

By the condition of (15), we have A ' 1 k1/2(logk)1/3

lkk=1klPi lm

A ' (logk)1/s

<kl3/2plh t1/2

<cyilogr y_L < cf

- JL 13/2 - /L 1-1/2- L^

■(log I )U371/2

1 l3/2Pl t,k1/2~ =1 I3'2Pi

(log o1'3

yy—-

o((logn)2(loglogn)-ß), (39)

n i i-1 i n i

< f (log I)

O ((log n) (log log n)

By Chebyshev's inequality and the condition of (15) and (35), we obtain

<y Var (Sk) < y J_ y

< L^ y 2 < Cf ¡Pi L -(log-)2'3

h lPi k=1 K 1=1

hi lPi

Hence (37)-(40) imply that n 1-1 Cov (,

<cf ((°g^=o((logn)2(loglogn)-ß

i=1 k=1

(ak,ai)

o((logn)2(loglogn)-ß). (41)

But Var(ak) = 0 if pk = 0 and

Var (ak) = X—PPk < — if pk = 0.

£ Var (ak)

£ 1(2

< £ _L < £ (l°gfc)

1<k<nk2pk 1<k<n kpk

Pk =0 pt = 0

= o ((log

Equations (41) and (43) together imply that Var (Tl7 )=0 ((logn)2(log log n)-p) , as

n ^ X. (44)

Hence applying Remark 7, our theorem is proved under the restricting condition (36).

Nowwe drop the restricting condition (36). Fixx > 0 and define

c?k = max (ak, -xak1/2) , bk = min (bk, xak1'2) ,

Pk = P(*k <Sk <bk),

where a is defined by (3).

Clearly pk < pk, and so assuming pk = 0; then, also we have pk = 0, and thus

ak = -1l{ak <Sk < hk} pk

= -^(i[sk <Sk <bk} pk

+I{ak <Sk <ak} + l{bk <Sk <bk})

< --I {ak <Sk < bk} pk

+ -1(l {ak <Sk <ak} + I [bk <Sk < bk}) pk

< --I {ak <Sk < \} pk

l{sk <-xak1/2} l{sk >xak1/2}

we obtain

lim PÎ-xak1'2 < Sk < 0)

= lim P (-xak <Sk < 0) = O (0) - O (-x),

k —ï ¡.y:)

lim P(0< Sk < xak1/2)

= lim P(0<Sk < xak) = O (x) - O (0).

Applying the almost sure central limit theorem for NA random variables (5), that is,

lim :-£-1 {Sfc < xak112} = O (x) a.s. Vx e R,

1 log nk=1k

Lemma 8, and (48), we have

1 " I {Sk < -xak1'2}

lognf=lkP (-xok1/2 <Sk <0) O(-x)

0(0)-0(-x) 1 n l{Sk > xak1/2}

lim / , .,

logn k='1kP (0 <Sk < xak112)

1 - O(x) = O (x) - O (0)

Since âk and bk satisfy (36), we get

1 £ak

lim i-£-r

log n k

= 1 a.s.,

I [dk <Sk < bk} pk

if Pk =0,

if Pk = 0.

Equations (46) and (50)-(51) together imply that

1 ^«k 1 - O (x)

lim sup -- }-r <1 + 2———H—

logn^k O(x)-O(0)

On the other hand, if pk = 0, then we have

I{ak <Sk <bk\

P(-xak1/2 <Sk <0) P(0<Sk <xak1/2Y >±l[ak < Sk <bk}Î1-Pk - Pk )

. . ..... „ ... . Pk V Pk )

By (35) and the central limit theorem for NA random variables (4), that is,

P(^ < x ) - O (x)

> ak\1 -

= o(1),

a.s. (53)

P (Sk < -axk1/2) + P(sk > oxk1/2) \

minjP(0 <Sk < oxk1/2),P(-oxk1/2 < Sk < 0)}J,

and by the central limit theorem,

P (Sk < -axk1/2) + P(sk > oxk1/2) k^ min{P(0 <Sk < axk1/2),P(-axk1/2 <Sk < 0)} 1 - O (x)

= 1-2-

0(x)-0(Q)'

Applying Lemma 8, (51) and (54) imply that

,. • r 1 v1 ak l-O(x)

lim inf -- y -k >1- 2-i-— a.s., (56)

n^™ logn k= k 0(x)-0(0) K '

and hence

1-®(x) > lim sup 0(x)-0(0) logn k=lk

1 n ak

> lim inf --Y

logn k=1 fc

>1-2- a.s.

O(%) - O(0)

By the arbitrariness of x, let x ^ to in (57); we have

, ,. 1 ^rak

1> lim sup--Yt

n^tx logn = k

> lim inf- Y >1 a.s.

logn k

1 v1 ak

lim -- y -k = 1 a.s.

logn k

This completes the proof of Theorem 2.

Proof of Theorem 5. Let k <1 - T, 1 < k < I, and et = l3a'2; we have

Cov (ak, ai)

< —— (p (ai -bk <Si - Sk+i« + Sk+i« - Sk Pi Pk

< bi -ak,ak <Sk < bk)-PiPk)

< —— (p (ai -bk <Si - Sk+i* + Sk+i* - Sk Pi Pk

<bi -ak)p(ak <Sk <bk)-PiPk) (60)

<—(p (ai - bk - ei <Si - Sk+i- <bi -ak + Pi

-Pi + p(\Sk+r -Sk| > ei))

<—(p (ai - bk - ei < Si-k-i* <bi -ak + Pi

-Pi + p(\SP\> ei)).

Applying Lemma 9, (33), and (35), we obtain P (ax -bk - ei < Si_k_i* <k -fyk + ti) - pi

)-o( &))

( ' i1'2> U-k-i«f2

i1/5 (i-k-n1/5 Hence Lemma 11, (60), and (61) imply that

1<k<i<n k<i-ia

< k< i k< i- i

On the other hand, y

1< k< i< n k>i-ia

Cov (ak, ai k

= O (logn).

k, ai)

= O (logn),

because I - la < k < I, F -(I- la)y ^ 0 asl y < 1,a < 1.

But Var(ak) = 0 if pk = 0 and

Var (ak) = Izlk < _L if pk = 0.

(63) TO for

" Var (ak) v 1 A 1 fn^ < I ^Ik^ = O(logn). (65)

1< k< n

Pk = 0

Noting that

Var (I"!

Var (a

1< k< i< n k>i-ia

Cov (ak, a{t k

1< k< i< n k<i-ia

Cov (ak,ai k

thus (62)-(66) imply that

Var ( It )=O(logn),

as n —> to.

Hence applying Remark 7, we have

1 a lim -— , logn k=1 k

Ik = 1

This completes the proof of Theorem 5.

(68) □

Appendix

Proof of Lemma 6. Let hn = Yn=1 ^k^k, and then Ve > 0

Hn EHn Dn Dn

> e I <

Var (Hn/Dn

<c(logDn)-ß. (A.1)

Let r < 1,rß > 1, and n-k = inf[nk,Dnt > exp(kr)}, and then D > exp(kr), D. 1 < exp(kr), for Dn ~ Dn-i, we get

that is,

and thus

exp (kr) exp (kr)

Dnk ~ exp (kr),

exp (kr)

<1-^1,

Dnt_i exp ((k-1)r)

exp (kr

1 - (1 -

exp (kr

(r-kr-1).

On account of r < 1, then D„

exp (r -kr ^ —> 1, as k

Hnk EHnk

Dnt Dnt

< c£ -1-« < c£ -Ta

¿1 (log Dn)ß jLirß By the Borel-Cantelli lemma,

Dnt Dnt

< œ>.

0 a.s.

EHnk = IH1 dk = 1

Dnt = Dnt = '

1, a.s.

Now for nk-1 < n < nk, for Dn Î <x>, D /D Zk > 0, then Hn Î, and we have

(A.8) and by

Dnt-1 Hnt_1 ^ Hn

Dnt Dnt_

< — <

Hnk Dnk

Dn Dnt Dnt_

Hn D„

1 a.s.

This completes the proof of Lemma 6.

1 a.s. (A.9)

(A.10) □

Proof of Lemma 10. Applying Lemma 9, (33), and noting the conditions of (19) and 0 < a < 1/7,we get

Pk = P(*k <Sk <h)

<(O(&)-O(£ )) + <£

< (k1/2-a 1 ), V« <C(— +—5 )< C2k ■

(A.11)

Applying Lemma 9, (34), and noting the conditions of (19) and 0 < a < 1/7,we have

Pk = P("k <Sk <h)

>(o(A )-o(« ))i

\k1/2 \k1/2) k1/5

(A.12)

k1/2 k1/5

1 \ ^ u-«

+ 7T7T I > ^k .

Thus Lemma 10 immediately follows from (A.11) and (A.12).

Proof of Lemma 11. By Lemma 10, Chebyshev's inequality, and noting the condition of 0 < a < 1/7,we have

l<k<l<n KLPl k<l-l"

1 Var (V) n 1 r 1

< 1<£KnklPl k<l-l"

<C£ V+« £l (A.13)

l=1 1 k=1 K

<£ ^=0(.og»).

It proves (29). By Lemma 10 and 0 < a < 1/7,we get

¿-> 1-1 r,

1<k<i<nklPl (l-k-l«)1/5 k<l-l"

<c£ TT« ( £

1 l1« \1<k<(l-la)/2k(l -la -k)

<c£ n~«

(l-l«)/2<k<l-pk(l -l« -k)

1 £ 1

1<k<(l-la)/2

(A.14)

, . - £ 1

11-«\ (i-i« )1/5

— £ —

\-l« ¿^ 1-1/5 1 1 1<k<(l-la )/2K

n log n 1

<c£ j-àns <£ = o(log,

l=11 l=11

It proves (30). Applying Lemma 9, (33), (35), el = l3a/2, and noting the condition of 0 < a < 1/7,we obtain

1<k<l<n klPl k<l-l"

(l-k- n1'2) \l1/2J 1

f I^L ( 1

< C 1<fl<nklPl ((l-k-H'2 <

(A.15)

i<Ùl<nklPl (l-k-n1'2 k<l-l"

1<tl<nklPl (l-k-n1'2

:— + + I3.

Now applying the same procedure as before, we have

V -al I 1 k 1 <c f -¡à(-- +

i<k<l<nklPl\i1-ad-k-ia)in id-k-ia)1/2 k<l-l"

<cf W1-

A 1-1/2

(l ]a)i'2 k I-la ^ ki'2

(l - 1 ) 1<k<(l-la)/2 * 1 1 1<k<(l-la)/2 *

f 13/2 f

1< l< n

13/21<ÙÎ-1« (l-k-n1/2

«f ^ <cf 11=0(logn),

12-a. — " L-, 1 1< l< n 1< l< n

12 <C f

1<k<l<n l1-ak1/2+a(l - I« -k)1'2

k<l-l"

1< l< n

1 f J_

+ ( 1 ia)1/2+<* f k.1'2 (l-1 ) 1<k<(l-la )/2li

1< l< n

1 I 1 l1/2-a +

l1-<*\ (¡-I-)1/2 (l-la)

^ ('-nm )

<cf J=0(logn).

1< l< n

(A.16)

Noting that 0 < a < 1/7,we deduce 1

1 <c f

1<£Kni1-(7/2)ak1/2(i-ia -k)1/2 k<l-l"

¡1-(7/2)a

1< l< n

1 ^ 1 1 x\ -7TT f T +

^ . f - )

( 1 ia)1/2 ^ k (I -la) ^ k1'2 I

(i-i ) 1<k<(l-la)/2K ( 1 ) 1<k<(l-l")/2K /

<cf1-=0(logn).

l3/2_(7/2)a ^ " L-. 1 1<i<n 1<i<n

(A.17)

It proves (31). The proof of (32) is similar to the proof of (31). This completes the proof of Lemma 11. □

Acknowledgments

The authors are very grateful to the academic editor, professor Ying Hu, and the two anonymous reviewers for their valuable comments and helpful suggestions, which significantly contributed to improving the quality of this paper. This work is jointly supported by National Natural Science Foundation of China (11061012,71271210), Project Supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ((2011)47), the Guangxi China Science Foundation (2013GXNSFDA019001).

References

[1] K. Joag-Dev and F. Proschan, "Negative association of random variables, with applications," The Annals of Statistics, vol. 11, no.

1, pp. 286-295,1983.

[2] Q. M. Shao, "A comparison theorem on moment inequalities between negatively associated and independent random variables," Journal of Theoretical Probability, vol. 13, no. 2, pp. 343356, 2000.

[3] Q. Y. Wu and Y. Y. Jiang, "A law of the iterated logarithm of partial sums for NA random variables," Journal of the Korean Statistical Society, vol. 39, no. 2, pp. 199-206, 2010.

[4] C. M. Newman, "Asymptotic independence and limit theorems for positively and negatively dependent random variables," in Inequalities in Statistics and Probability, vol. 5 of Lecture Notes-Monograph Series, pp. 127-140, Institute of Mathematical Statistics, Hay ward, Calif, USA, 1984.

[5] P. Matula, "Some limit theorems for negatively dependent random variables," Yokohama Mathematical Journal, vol. 41, no.

2, pp. 163-173,1994.

[6] P. Matula, "On almost sure limit theorems for positively dependent random variables," Statistics & Probability Letters, vol. 74, no. 1, pp. 59-66, 2005.

[7] Q. Y. Wu, "An almost sure central limit theorem for the weight function sequences of NA random variables," Proceedings— Mathematical Sciences, vol. 121, no. 3, pp. 369-377, 2011.

[8] G. A. Brosamler, "An almost everywhere central limit theorem," Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 3, pp. 561-574, 1988.

[9] P. Schatte, "On strong versions of the central limit theorem," Mathematische Nachrichten, vol. 137, pp. 249-256,1988.

[10] M. Peligrad and Q. M. Shao, "A note on the almost sure central limit theorem for weakly dependent random variables," Statistics & Probability Letters, vol. 22, no. 2, pp. 131-136,1995.

[11] P. Matula, "On the almost sure central limit theorem for associated random variables," Probability and Mathematical Statistics, vol. 18, no. 2, pp. 411-416, 1998.

[12] E. Csaki, A. Foldes, and P. Revesz, "On almost sure local and global central limit theorems," Probability Theory and Related Fields, vol. 97, no. 3, pp. 321-337,1993.

[13] G. Hurelbaatar, "An almost sure local and global central limit theorem for weakly dependent random variables," Annales Universitatis Scientiarum Budapestinensis de Rolando Eotvos Nominatae, Sectio Mathematica, vol. 38, pp. 109-126,1995.

[14] Z. C. Weng, Z. C. Peng, and S. Nadarajah, "The almost sure local central limit theorem for the product of partial sums," Proceedings—Mathematical Sciences, vol. 121, no. 2, pp. 217-228, 2011.

[15] I. Berkes and E. Csaki, "A universal result in almost sure central limit theory," Stochastic Processes and their Applications, vol. 94, no. 1, pp. 105-134, 2001.

[16] A. Foldes, "Some of my favorite results with Endre Csaki and Pal Ravesz: a survey," Periodica Mathematica Hungarica, vol. 50, no. 1-2, pp. 117-134, 2005.

[17] J. M. Pan, "On the convergence rate in the central limit theorem for negatively associated sequences," Chinese Journal of Applied Probability and Statistics, vol. 13, no. 2, pp. 183-192,1997.

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