2. Strong Law of Large Numbers and Growth Rate for AANA Sequence

Volume 2010, Article ID 218380,15pages doi:10.1155/2010/218380

Research Article

Convergence Properties for Asymptotically almost Negatively Associated Sequence

Xuejun Wang, Shuhe Hu, and Wenzhi Yang

School of Mathematical Science, Anhui University, Hefei 230039, China

Correspondence should be addressed to Shuhe Hu,[email protected] Received 20 July 2010; Revised 9 October 2010; Accepted 2 November 2010 Academic Editor: Ibrahim Yalcinkaya

Copyrightq2010 Xuejun Wang et al. 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.

We get the strong law of large numbers, strong growth rate, and the integrability of supremum for the partial sums of asymptotically almost negatively associated sequence. In addition, the complete convergence for weighted sums of asymptotically almost negatively associated sequences is also studied.

1. Introduction

Definition 1.1. A finite collection of random variablesX₁, X₂, . . . , X_n is said to be negatively associatedNAif, for every pair of disjoint subsetsA1,A2of{1,2, . . . , n},

Cov

fXi:i∈A1, g

Xj :j∈A2

≤0, 1.1

wheneverf and g are coordinate-wise nondecreasing such that this covariance exists. An infinite sequence{Xn, n≥1}is NA if every finite subcollection is NA.

The concept of negative association was introduced by Joag-Dev and Proschan1and Block et al.2. By inspecting the proof of maximal inequality for the NA random variables in Matuła 3, one also can allow negative correlations provided they are small. Primarily motivated by this, Chandra and Ghosal4,5introduced the following dependence.

Definition 1.2. A sequence{Xn, n ≥ 1}of random variables is called asymptotically almost negatively associatedAANAif there exists a nonnegative sequenceqn → 0 asn → ∞ such that

Cov

fXn, gXn1, X_n2, . . . , X_nk

≤qn Var

fXn Var

gXn1, X_n2, . . . , X_nk1/2 1.2

for all n, k ≥ 1 and for all coordinate-wise nondecreasing continuous functions f and g whenever the variances exist.

The family of AANA sequence contains NAin particular, independent sequences withqn 0,n ≥ 1and some more sequences of random variables which are not much deviated from being negatively associated. An example of an AANA sequence which is not NA was constructed by Chandra and Ghosal4.

Since the concept of AANA sequence was introduced by Chandra and Ghosal 4, many applications have been found. For example, Chandra and Ghosal 4 derived the Kolmogorov-type inequality and the strong law of large numbers of Marcinkiewicz- Zygmund, Chandra and Ghosal 5 obtained the almost sure convergence of weighted averages, Ko et al. 6 studied the H´ajek-R´enyi-type inequality, and Wang et al. 7 established the law of the iterated logarithm for product sums. Recently, Yuan and An8 established some Rosenthal-type inequalities for maximum partial sums of AANA sequence.

As applications of these inequalities, they derived some results on Lp convergence, where 1 < p < 2, and complete convergence. In addition, they estimated the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.

The main purpose of the paper is to study the strong law of large numbers, strong growth rate, and the integrability of supremum for AANA sequence. In addition, the complete convergence for weighted sums of AANA sequence is also studied.

Throughout the paper, we let{Xn, n≥ 1}be a sequence of AANA random variables defined on a fixed probability space Ω,F, P. Denote S_n . _n

i1X_i. LetX^a −aIX <

−a XI|X| ≤ a aIX > a for some a > 0, and let IA be the indicator function of the set A. For p > 1, let q . p/p − 1 be the dual number of p. We assume that φx is a positive increasing function on 0,∞ satisfying φx ↑ ∞ as x → ∞ and ψx is the inverse function of φx. Since φx ↑ ∞, it follows that ψx ↑ ∞. For easy notation, we let φ0 0 and ψ0 0. The a_n Obn denotes that there exists a positive constant C such that |an/b_n| ≤ C. C denotes a positive constant which may be different in various places. The main results of this paper are dependent on the following lemmas.

Lemma 1.3 cf. Yuan and An 8, Lemma 2.1. Let {Xn, n ≥ 1} be a sequence of AANA random variables with mixing coefficients{qn, n ≥ 1}, and letf1, f2, . . .be all nondecreasing (or nonincreasing) functions, then{fnXn, n≥1}is still a sequence of AANA random variables with mixing coefficients{qn, n≥1}.

Lemma 1.4. Let 1 < p ≤ 2, and let {Xn, n ≥ 1} be a sequence of AANA random variables with mixing coefficients {qn, n ≥ 1} and EX_n 0 for each n ≥ 1. If _∞

n1q²n < ∞, then there exists a positive constant C_p depending only on p such that

E max

1≤i≤n|Si|^p

≤C_p n

i1

E|Xi|^p 1.3

for alln≥1, whereC_p2^p2^2−pp 6p^p_∞

n1q²n^p/q.

Proof. We use the same notations as that in the study by Yuan and An8. They proved that

E max

1≤i≤nSi

^p≤2^2−pp n

i1

Xi^pp

6p_pn−1

i1

q^2/qiXi_p p

,

E max

1≤i≤n−Si

^p≤2^2−pp n

i1

Xi^pp

6p_pn−1

i1

q^2/qiXi_p p

,

max1≤i≤n|Si|^p≤2^p−1 max

1≤i≤nSi

^p2^p−1 max

1≤i≤n−Si ^p.

1.4

By1.4and H ¨older’s inequality, we have E max

1≤i≤n|Si|^p

≤2^p−1E max

1≤i≤nSi

^p2^p−1E max

1≤i≤n−Si ^p

≤2^p

2^2−pp n i1

E|Xi|^p

6p_{p n}

i1

q^2/qiXi_{p p}

≤2^p

⎡

⎣2^2−pp n

i1

E|Xi|^p 6pp

_n

i1

q²i

_{p/q n}

i1

E|Xi|^p

⎤

⎦

≤2^p

⎡

⎣2^2−pp 6pp

_∞

n1

q²n p/q⎤

⎦ⁿ

i1

E|Xi|^pC_p n

i1

E|Xi|^p.

1.5

This completes the proof of the lemma.

We point out thatLemma 1.4has been studied by Yuan and An8. But here we give the accurate coefficientC_p. AndLemma 1.4generalizes and improves the result of Lemma 2.2 in the study by Ko et al.6.

Lemma 1.5cf. Fazekas and Klesov9, Theorem 2.1 and Hu et al.10, Lemma 1.5. Let {Xn, n≥1}be a sequence of random variables. Letb₁, b₂, . . .be a nondecreasing unbounded sequence of positive numbers, and letα₁, α₂, . . .be nonnegative numbers. LetrandCbe fixed positive numbers.

Assume that, for eachn≥1,

E max

1≤l≤n|Sl|^r

≤C n

l1

α_l, 1.6

∞ l1

α_l

b^r_l <∞, 1.7

then

nlim→ ∞

Sn

b_n 0 a.s., 1.8

and with the growth rate

S_n

bn O β_n bn

a.s., 1.9

where

β_n max

1≤k≤nb_kv_k^δ/r, ∀0< δ <1, v_n^∞

kn

α_k

b^r_k, lim

n→ ∞

β_n bn 0, E max

1≤l≤n

Sl

b_l ^r

≤4C n

l1

αl

b^r_l <∞, E

sup

l≥1

S_l bl

^r

≤4C ∞

l1

α_l b^r_l <∞.

1.10

If further one assumes thatα_n >0 for infinitely manyn, then

E

sup

l≥1

S_l β_l

^r

≤4C ∞

l1

α_l

β^r_l <∞. 1.11

Lemma 1.6 cf. Fazekas and Klesov 9, Corollary 2.1 and Hu 11, Corollary 2.1.1.

Let b₁, b₂, . . . be a nondecreasing unbounded sequence of positive numbers, and let α₁, α₂, . . . be nonnegative numbers. DenoteΛk α1α2· · ·αkfork ≥ 1. Letr be a fixed positive number satisfying1.6. If

∞ l1

Λl

1 b^r_l − 1

b^r_l1

<∞, 1.12

Λn

b^r_n is bounded, 1.13

then1.8–1.11hold.

Lemma 1.7cf. Yuan and An8, Theorem 2.1. Let{Xn, n≥1}be a sequence of AANA random variables withEXi 0 for all i ≥ 1 andp ∈ 3·2^k−1,4·2^k−1, where integer numberk ≥ 1. If _∞

n1q^q/pn < ∞, then there exists a positive constantD_p depending only onp such that, for all n≥1,

E max

1≤i≤n|Si|^p

≤D_p

⎧⎨

⎩ n

i1

E|Xi|^p _n

i1

EX²_{i p/2}⎫

⎬

⎭. 1.14

Lemma 1.8. Assume that the inverse functionψxofφxsatisfies

ψnⁿ

i1

1

ψi On. 1.15

IfEφ|X|<∞, then_∞

n11/ψnE|X|I|X|> ψn<∞.

Proof. Sinceψxis an increasing function ofx, we have that ∞

n1

1

ψnE|X|I

|X|> ψn ^∞

n1

1 ψn

∞ in

E|X|I

ψi<|X| ≤ψi1

^∞

i1

E|X|I

ψi<|X| ≤ψi1ⁱ

n1

1 ψn

≤^∞

i1

P

ψi<|X| ≤ψi1

ψi1ⁱ

n1

1 ψn

≤C ∞

i1

P

ψi<|X| ≤ψi1 i

≤CE φ|X|

<∞.

1.16

The proof is complete.

2. Strong Law of Large Numbers and Growth Rate for AANA Sequence

Theorem 2.1. Let {Xn, n ≥ 1} be a sequence of mean zero AANA random variables with _∞

n1q²n <∞, and let{bn, n≥ 1}be a nondecreasing unbounded sequence of positive numbers;

1< p≤2. Assume that

∞ n1

E|Xn|^p

b^p_n <∞, 2.1

then

nlim→ ∞

S_n

bn 0 a.s., 2.2

and with the growth rate

S_n

b_n O β_n b_n

a.s., 2.3

where

β_nmax

1≤k≤nb_kv_k^δ/2, ∀0< δ <1, v_n^∞

kn

α_k

b^p_k, lim

n→ ∞

β_n bn 0, α_kC_pE|Xk|^p, k≥1, C_p is defined in Lemma 1.4,

E max

1≤l≤n

S_l b_l

^p

≤4 n l1

α_l b^p_l <∞, E

sup

l≥1

Sl

b_l ^p

≤4 ∞

l1

αl

b^p_l <∞.

2.4

If further one assumes thatα_n >0 for infinitely manyn, then

E

sup

l≥1

Sl

β_l ^p

≤4 ∞

l1

αl

β^p_l <∞. 2.5

Proof. ByLemma 1.4, we have

E max

1≤k≤n|Sk|^p

≤C_p n k1

E|Xk|^pn

k1

α_k. 2.6

It follows from2.1that

∞ n1

α_n b^p_n Cp

∞ n1

E|Xn|^p

b^p_n <∞. 2.7

Thus,2.2–2.5follow from2.6,2.7, andLemma 1.5immediately. We complete the proof of the theorem.

Theorem 2.2. Let{Xn, n ≥ 1}be a sequence of AANA random variables with_∞

n1q²n < ∞, 1≤p <2. DenoteQ_nmax_1≤k≤nEX_k²forn≥1 andQ₀0. Assume that

∞ n1

Q_n

n^2/p <∞, 2.8

then

nlim→ ∞

1 n^1/p

n i1

Xi−EXi 0 a.s., 2.9

and with the growth rate

1 n^1/p

n i1

Xi−EXi O βn

n^1/p

a.s., 2.10

where

β_n max

1≤k≤nk^1/pv^δ/2_k , ∀0< δ <1, v_n^∞

kn

α_k

k^2/p, lim

n→ ∞

β_n n^1/p 0, αkC2kQk−k−1Qk−1, k≥1, C2 is defined in Lemma 1.4,

2.11

E

max1≤l≤n

Sl

l^{1/p 2}

≤4 n

l1

αl

l^2/p <∞, 2.12

E

sup

l≥1

S_l l^1/p

²

≤4 ∞

l1

α_l

l^2/p <∞. 2.13

If further one assumes thatα_n >0 for infinitely manyn, then

E

sup

l≥1

S_l β_l

²

≤4 ∞

l1

α_l

β²_l <∞. 2.14

In addition, for anyr∈0,2,

E

sup

l≥1

S_l l^1/p

^r

≤1 4r 2−r

∞ l1

α_l

l^2/p <∞. 2.15

Proof. Assume thatEX_n 0,b_n n^1/p, andΛn _n

l1α_l, n ≥ 1. ByLemma 1.4, we can see that

E

⎛

⎝max

1≤k≤n

k i1

X_i

2⎞

⎠≤C₂ n

i1

EX_i²≤C₂nQ_n ⁿ

k1

α_k. 2.16

It is a simple fact thatαk≥0 for allk≥1. It follows from2.8that ∞

l1

Λl

1 b²_l − 1

b²_l1

C2

∞ l1

lQl

1

l^2/p − 1 l1^2/p

≤ 2C₂ p

∞ l1

Q_l

l^2/p <∞. 2.17

That is to say that1.12holds. By Remark 2.1 in Fazekas and Klesov9,1.12implies1.13.

ByLemma 1.6, we can obtain2.9–2.14immediately. By2.13, it follows that

E

sup

l≥1

S_l l^1/p

^r

_∞

0

P

sup

l≥1

S_l l^1/p

^r > t

dt≤1 _∞

1

P

sup

l≥1

S_l l^1/p

> t^1/r

dt

≤1E

sup

l≥1

S_l l^1/p

^{2 ∞}

1

t^−2/rdt≤1 4r 2−r

∞ l1

α_l l^2/p <∞.

2.18

The proof is complete.

Theorem 2.3. Let p ∈ 3·2^k−1,4·2^k−1, where integer numberk ≥ 1, and let {Xn, n ≥ 1} be a sequence of AANA random variables with EXi 0 for all i ≥ 1 and _∞

n1q^q/pn < ∞. Let {bn, n≥1}be a nondecreasing unbounded sequence of positive numbers. Assume that

∞ n1

n^p/2−1

b^p_n E|Xn|^p<∞, ∞

k1

E|Xk|^{p ∞}

nk1

n^p/2−2 b^p_n <∞,

2.19

then1.8–1.11hold (forC1), where

α₁2D_pE|X1|^p, α_k2D_p

⎛

⎝k^p/2−1 k j1

EX_j^p−k−1^p/2−1k−1

j1

EX_j^p⎞

⎠, k≥2, 2.20

andD_pis defined inLemma 1.7.

Proof. Sincep >2, 0<2/p <1. ByCr’s inequality, _n

i1

|Xi|^p _2/p

≤ⁿ

i1

X_i², 2.21

which implies that

n i1

E|Xi|^p≤E _n

i1

X_i² _p/2

. 2.22

By Jensen’s inequality, we have _n

i1

EX_i² _p/2

≤E _n

i1

X_i² _p/2

. 2.23

By2.22-2.23andCr’s inequality,

n i1

E|Xi|^p _n

i1

EX_i² _p/2

≤2E _n

i1

X_i² _p/2

≤2n^p/2−1 n

i1

E|Xi|^p. 2.24

It follows fromLemma 1.7and2.24that

E max

1≤i≤n|Si|^p

≤2D_pn^p/2−1 n

i1

E|Xi|^pn

l1

α_l. 2.25

It is a simple fact that

0≤α_k≤C₁ p⎛

⎝k^p/2−1E|Xk|^pk^p/2−2 k−1

j1

EX_j^p⎞

⎠, 2.26

whereC₁pis a positive number depending only onpandD_p. By2.19,

∞ n1

αn

b^p_n ≤C1

p_∞

n1

n^p/2−1

b^p_n E|Xn|^p∞

k1

E|Xk|^{p ∞}

nk1

n^p/2−2 b^p_n

<∞. 2.27

The desired results follow from2.25–2.27andLemma 1.5immediately.

3. Complete Convergence for Weighted Sums of AANA Random Variables

Theorem 3.1. Let{X, Xn, n≥1}be a sequence of identically distributed AANA random variables with_∞

n1q²n <∞,EX 0,EX² < ∞, andEφ|X| <∞. Assume that the inverse function ψxofφxsatisfies1.15. Let{ani, n≥1, i≥1}be a triangular array of positive constants such that

imax_1≤i≤na_niO1/ψn,

ii_n

i1a²_niOlog^−1−αnfor someα >0.

Then, for anyε >0,

∞ n1

n⁻¹P

max1≤j≤n

j i1

aniXi

> ε

<∞. 3.1

Proof. For eachn≥1, denote

X_jⁿ−ψnI

Xj <−ψn

XjIXj≤ψn

ψnI

Xj> ψn

, 1≤j ≤n, T_jⁿ

j i1

a_niX_iⁿ−Ea_niX_iⁿ!

, 1≤j≤n,

A"ⁿ

i1

X_iX_iⁿ! "ⁿ

i1

|Xi| ≤ψn

, BA#ⁿ

i1

X_i/X_iⁿ! #ⁿ

i1

|Xi|> ψn ,

E_n

max1≤j≤n

j i1

a_niX_i > ε

.

3.2

It is easy to check that

j i1

a_niX_iT_jⁿ j

i1

Ea_niX_iⁿ j

i1

a_niX_iI

|Xi|> ψn

^j

i1

aniψn I

Xj<−ψn

−I

Xj> ψn ,

EnEnAEnB

max1≤j≤n

T_jⁿ

j i1

EaniXⁿ_i > ε

EnB

⊂

max1≤j≤n

T_jⁿ> ε−max

1≤j≤n

j i1

EaniX_iⁿ

B.

3.3

Therefore,

PEn≤P

max1≤j≤n

T_jⁿ> ε−max

1≤j≤n

j i1

EaniX_iⁿ

PB

≤P

max1≤j≤n

T_jⁿ> ε−max

1≤j≤n

j i1

EaniXⁿ_i

ⁿ

i1

P

|Xi|> ψn .

3.4

Firstly, we will show that

max1≤j≤n

j i1

EaniXⁿ_i

−→0, asn−→ ∞. 3.5

It follows fromLemma 1.8and Kronecker’s lemma that 1

ψn n

i1

E|X|I

|X|> ψi

−→0, asn−→ ∞. 3.6

ByEX0, conditioni,3.6, andψn↑ ∞, we can see that

max1≤j≤n

j i1

EaniX_iⁿ ≤ⁿ

i1

EaniXiI

|Xi| ≤ψnⁿ

i1

aniψnEI

|Xi|> ψn

≤ⁿ

i1

aniE|Xi|I

|Xi|> ψn ⁿ

i1

aniE|Xi|I

|Xi|> ψn

≤ C ψn

n i1

E|X|I

|X|> ψi

−→0, asn−→ ∞,

3.7

which implies3.5. By3.4and3.5, we can see that, for sufficiently largen,

P

max1≤j≤n

j i1

a_niX_i > ε

≤P max

1≤j≤n

T_jⁿ> ε 2

ⁿ

i1

P

|Xi|> ψn

. 3.8

To prove3.1, it suffices to show that ∞ n1

n⁻¹P max

1≤j≤n

T_jⁿ> ε 2

<∞,

∞ n1

n⁻¹ n

i1

P

|Xi|> ψn

<∞.

3.9

By Markov’s inequality,Lemma 1.4,Cr inequality,EX²<∞, and conditionii, we have ∞

n1

n⁻¹P max

1≤j≤n

T_jⁿ> ε 2

≤C ∞ n1

n⁻¹E max

1≤j≤n

T_jⁿ²

≤C ∞ n1

n⁻¹ n

i1

EaniX_iⁿ²

≤C ∞ n1

n⁻¹ n

i1

a²_niEX²I

|X| ≤ψn C

∞ n1

n⁻¹ n i1

a²_niψ²nEI

|X|> ψn

≤C ∞ n1

n⁻¹ n

i1

a²_niEX²I

|X| ≤ψn C

∞ n1

n⁻¹ n i1

a²_niEX²I

|X|> ψn

≤C ∞ n1

n⁻¹ n

i1

a²_ni ≤C ∞ n1

n⁻¹log^−1−αn <∞.

3.10

It follows fromEφ|X|<∞that ∞

n1

n⁻¹ n

i1

P

|Xi|> ψn ^∞

n1

P

|X|> ψn ^∞

n1

P

φ|X|> n

≤CE φ|X|

<∞.

3.11

Theorem 3.2. Let{Xn, n≥1}be a sequence of AANA random variables, and let{ani, n≥1, i≥1}

be an array of positive numbers. Let{bn, n≥ 1}be an increasing sequence of positive integers, and let{cn, n ≥1}be a sequence of positive numbers. If, for somep ∈3·2^k−1,4·2^k−1, where integer numberk≥1, 0< t <2, and for anyε >0, the following conditions are satisfied:

∞ n1

c_n

bn

i1

P |aniX_i| ≥εb^1/t_n !

<∞,

∞ n1

c_nb^−p/t_n

bn

i1

|ani|^pE|Xi|^pI |aniX_i|< εb^1/t_n !

<∞,

∞ n1

c_nb_n^−p/t _b

n

i1

a²_niEX_i²I |aniX_i|< εb^1/t_n !^p/2

<∞,

3.12

and_∞

n1q^q/pn<∞, then

∞ n1

c_nP

⎧⎨

⎩max

1≤i≤bn

i j1

$

a_njX_j−a_njEX_jI a_njX_j< εb_n^1/t!%

≥εb_n^1/t

⎫⎬

⎭<∞. 3.13 Proof. Note that if the series _∞

n1c_n is convergent, then 3.13 holds. Therefore, we will consider only such sequences{cn, n≥1}for which the series_∞

n1cnis divergent.

Let

Y_iⁿ−εb^1/t_n I aniXi<−εb_n^1/t!

aniXiI |aniXi|< εb_n^1/t!

εb^1/t_n I aniXi> εb^1/t_n ! ,

S_ni ⁱ

j1

Y_jⁿ, n≥1, i≥1.

3.14

Note that

P

⎧⎨

⎩max

1≤i≤bn

i j1

$

a_njX_nj−a_njEX_jI a_njX_j< εb^1/t_n !%

≥εb^1/t_n

⎫⎬

⎭

≤C

bn

i1

P |aniXi| ≥εb^1/t_n !

2^pε^−pb^−p/t_n E max

1≤i≤bn

|S_ni−ES_ni| p

.

3.15

Using the C_r inequality and Jensen’s inequality, we can estimate E|Y_iⁿ−EY_iⁿ|^p in the following way:

EY_iⁿ−EY_i^np≤C|ani|^pE|Xi|^pI |aniXi|< εb_n^1/t!

Cb_n^p/tP |aniXi| ≥εb^1/t_n !

. 3.16

By3.15,3.16, andLemma 1.7, we can get

P

⎧⎨

⎩max

1≤i≤bn

i j1

$

anjXj−anjEXjI anjXj< εb^1/t_n !%

≥εb^1/t_n

⎫⎬

⎭

≤C

bn

i1

P |aniXi| ≥εb^1/t_n !

Cb^−p/t_n

bn

i1

|ani|^pE|Xi|^pI |aniXi|< εb_n^1/t! Cb^−p/t_n

_b n

i1

a²_niEX_i²I |aniX_i|< εb^1/t_n !^p/2 .

3.17

Therefore, we can conclude that3.13holds by3.12and3.17.

Theorem 3.3. Let 1 ≤ r ≤ 2 and let{Xn, n ≥ 1}be a sequence of AANA random variables with EX_n 0 andE|Xn|^r <∞forn≥1. Let{ani, n≥1, i≥1}be an array of real numbers satisfying the condition

n i1

|ani|^rE|Xi|^r O n^δ!

asn−→ ∞ 3.18

and_∞

n1q^q/pn<∞for some 0< δ≤2/pandp∈3·2^k−1,4·2^k−1, where integer numberk≥1.

Then, for anyε >0 andαr≥1,

∞ n1

n^αr−2P

⎛

⎝max

1≤i≤n

i j1

anjXj

≥εn^α

⎞

⎠<∞. 3.19

Proof. Takecnn^αr−2,bnn, and 1/tαinTheorem 3.2. By3.18, we have ∞

n1

c_n

bn

i1

P |aniX_i| ≥εb^1/t_n !

≤C ∞ n1

n^αr−2 n

i1

|ani|^rE|Xi|^r n^αr ≤C

∞ n1

n^−2δ<∞,

∞ n1

c_nb^−p/t_n

bn

i1

|ani|^pE|Xi|^pI |aniX_i|< εb_n^1/t!

≤^∞

n1

n⁻² n

i1

|ani|^rE|Xi|^r ≤C ∞ n1

n^−2δ<∞,

∞ n1

c_nb_n^−p/t _b

n

i1

a²_niEX²_iI |aniX_i|< εb^1/t_n !^p/2

≤C ∞ n1

n^{αr−2−αrp/2} _n

i1

|ani|^rE|Xi|^r _p/2

≤C ∞ n1

nαr−2−αrp/2δp/2≤C ∞ n1

n^{αr1−p/2−1} <∞ 3.20

following fromδp/2≤1. By the assumptionEXn0 forn≥1 and3.18, we get 1

n^αmax

1≤i≤n

i j1

a_njEX_jIa_njX_j< εn^α

≤ 1 n^α

n j1

anjEXjIanjXj< εn^α 1 n^α

n j1

anjEXjIanjXj≥εn^α

≤ 1 n^αr

n j1

anj^rEXj^r ≤Cn^δ−αr −→0, asn−→ ∞

3.21

following fromδ < 1 andαr ≥ 1. We get the desired result fromTheorem 3.2immediately.

The proof is complete.

Theorem 3.4. Let{Xn, n≥1}be a sequence of AANA random variables satisfying_∞

n1q²n<∞, and let{ani, n≥1, i≥1}be an array of positive numbers. Let{bn, n≥1}be an increasing sequence of positive integers, and let{cn, n ≥ 1} be a sequence of positive numbers. If, for some 1< p ≤ 2, 0< t <2, and for anyε >0, the following conditions are satisfied:

∞ n1

c_n

bn

i1

P |aniX_i| ≥εb^1/t_n !

<∞,

∞ n1

c_nb^−p/t_n

bn

i1

|ani|^pE|Xi|^pI |aniX_i|< εb^1/t_n !

<∞,

3.22

then

∞ n1

c_nP

⎧⎨

⎩max

1≤i≤bn

i j1

$

a_njX_j−a_njEX_jI a_njX_j< εb_n^1/t!%

≥εb_n^1/t

⎫⎬

⎭<∞. 3.23 Proof. The proof is similar to that ofTheorem 3.2, so we omit it.

Acknowledgments

The authors are most grateful to the Editor Ibrahim Yalcinkaya and anonymous referee for careful reading of the manuscript and valuable suggestions, which helped to improve an earlier version of this paper. This paper was supported by the NNSF of ChinaGrant nos. 10871001, 61075009, Provincial Natural Science Research Project of Anhui Colleges KJ2010A005, Talents Youth Fund of Anhui Province Universities2010SQRL016ZD, Youth Science Research Fund of Anhui University2009QN011A, Academic innovation team of Anhui University KJTD001B, and Natural Science Research Project of Suzhou College 2009yzk25.

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