Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 717126, 11 pages
doi:10.1155/2011/717126
Research Article
Complete Convergence for Arrays of
Rowwise Asymptotically Almost Negatively
Associated Random Variables
Xuejun Wang, Shuhe Hu, and Wenzhi Yang
School of Mathematical Science, Anhui University, Hefei 230039, China
Correspondence should be addressed to Xuejun Wang, 07019@ahu.edu.cn
Received 24 April 2011; Revised 3 September 2011; Accepted 18 September 2011
Academic Editor: Wei-Der Chang
Copyright q 2011 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.
Let {Xni , i ≥ 1, n ≥ 1} be an array of rowwise asymptotically almost negatively associated
random variables. Some sufficient conditions for complete convergence for arrays of rowwise
asymptotically almost negatively associated random variables are presented without assumptions
of identical distribution. As an application, the Marcinkiewicz-Zygmund type strong law of large
numbers for weighted sums of asymptotically almost negatively associated random variables is
obtained.
1. Introduction
The concept of complete convergence was introduced by Hsu and Robbins 1 as follows.
A sequence of random variables {Un , n ≥ 1} is said to converge completely to a constant C if
∞
n1 P |Un − C| > ε < ∞ for all ε > 0. In view of the Borel-Cantelli lemma, this implies
that Un → C almost surely a.s.. The converse is true if the {Un , n ≥ 1} are independent.
Hsu and Robbins 1 proved that the sequence of arithmetic means of independent and
identically distributed i.i.d. random variables converges completely to the expected value
if the variance of the summands is finite. Since then many authors studied the complete
convergence for partial sums and weighted sums of random variables. The main purpose
of the present investigation is to provide the complete convergence results for weighted
sums of asymptotically almost negatively associated random variables and arrays of rowwise
asymptotically almost negatively associated random variables.
Firstly, let us recall the definitions of negatively associated and asymptotically almost
negatively associated random variables.
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Discrete Dynamics in Nature and Society
Definition 1.1. A finite collection of random variables X1 , X2 , . . . , Xn is said to be negatively
associated NA, in short if for every pair of disjoint subsets A1 , A2 of {1, 2, . . . , n},
Cov fXi : i ∈ A1 , g Xj : j ∈ A2 ≤ 0,
1.1
whenever f and g are coordinatewise nondecreasing such that this covariance exists. An
infinite sequence {Xn , n ≥ 1} is NA if every finite subcollection is negatively associated.
An array of random variables {Xni , i ≥ 1, n ≥ 1} is called rowwise NA random
variables if, for every n ≥ 1, {Xni , i ≥ 1} is a sequence of NA random variables.
The concept of negative association was introduced by Joag-Dev and Proschan 2. By
inspecting the proof of maximal inequality for the NA random variables in Matula 3, one
also can allow negative correlations provided they are small. Primarily motivated by this,
Chandra and Ghosal 4, 5 introduced the following dependence.
Definition 1.2. A sequence {Xn , n ≥ 1} of random variables is called asymptotically almost
negatively associated AANA, in short if there exists a nonnegative sequence qn → 0 as
n → ∞ such that
1/2
,
Cov fXn , gXn1 , Xn2 , . . . , Xnk ≤ qn Var fXn Var gXn1 , Xn2 , . . . , Xnk 1.2
for all n, k ≥ 1 and for all coordinatewise nondecreasing continuous functions f and g
whenever the variances exist.
An array of random variables {Xni , i ≥ 1, n ≥ 1} is called rowwise AANA random
variables if, for every n ≥ 1, {Xni , i ≥ 1} is a sequence of AANA random variables.
The family of AANA sequence contains NA in particular, independent sequences
with qn 0, n ≥ 1 and 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 Ghosal 4.
Since the concept of AANA sequence was introduced by Chandra and Ghosal 4,
many applications have been found. See, for example, Chandra and Ghosal 4 derived
the Kolmogorov type inequality and the strong law of large numbers of MarcinkiewiczZygmund, Chandra and Ghosal 5 obtained the almost sure convergence of weighted
averages, Ko et al. 6 studied the Hájek-Rényi type inequality, Wang et al. 7 established the
law of the iterated logarithm for product sums, Yuan and An 8 established some Rosenthal
type inequalities for maximum partial sums of AANA sequence, and Wang et al. 9 obtained
some strong growth rate and the integrability of supremum for the partial sums of AANA
random variables, and so forth.
Our goal in this paper is to study the complete convergence for arrays of
rowwise AANA random variables under some moment conditions. As an application,
the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of
AANA random variables is obtained. We will give some sufficient conditions for complete
convergence for an array of rowwise AANA random variables without assumption of
identical distribution. The results presented in this paper are obtained by using the truncated
method and the classical maximal type inequality of AANA random variables Lemma 1.5
below.
Discrete Dynamics in Nature and Society
3
Throughout the paper, let {Xni : i ≥ 1, n ≥ 1} be an array of rowwise AANA random
.
variables with the mixing coefficients {qi, i ≥ 1} in each row. For p > 1, let q p/p − 1
be the dual number of p. Let IA be the indicator function of the set A. C denotes a positive
constant which may be different in various places and an Obn stands for an ≤ Cbn .
Definition 1.3. An array of random variables {Xni , i ≥ 1, n ≥ 1} is said to be stochastically
dominated by a random variable X if there exists a positive constant C such that
P |Xni | > x ≤ CP |X| > x
1.3
for all x ≥ 0, i ≥ 1 and n ≥ 1.
The following lemmas are useful for the proofs of the main results.
Lemma 1.4 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}, let f1 , f2 , . . . be all nondecreasing (or all
nonincreasing) functions, then {fn Xn , n ≥ 1} is still a sequence of AANA random variables with
mixing coefficients {qn, n ≥ 1}.
Lemma 1.5 cf. Yuan and An 8, Theorem 2.1. Let {Xn , n ≥ 1} be a sequence of AANA random
variables with EXi 0 for all i ≥ 1 and p ∈ 3 · 2k−1 , 4 · 2k−1 , where integer number k ≥ 1. If
∞ q/p
n < ∞, then there exists a positive constant Dp depending only on p such that for all
n1 q
n ≥ 1,
j
p p/2 n
n
p
2
E|X
EX
.
E max
i1 Xi ≤ Dp
|
i
i
i1
i1
1≤j≤n
1.4
Lemma 1.6. Let {Xn , n ≥ 1} be a sequence of random variables which is stochastically dominated by
a random variable X. For any α > 0 and b > 0, the following two statements hold:
E|Xn |α I|Xn | ≤ b ≤ C1 E|X|α I|X| ≤ b bα P |X| > b ,
1.5
E|Xn |α I|Xn | > b ≤ C2 E|X|α I|X| > b,
1.6
where C1 and C2 are positive constants.
2. Main Results
Let {Xni : i ≥ 1, n ≥ 1} be an array of rowwise AANA random variables with the mixing
coefficients {qi, i ≥ 1} in each row, and let {ani : i ≥ 1, n ≥ 1} be an array of real numbers. Let
{Xi , i ≥ 1} be a sequence of AANA random variables with the mixing coefficients {qi, i ≥ 1}
and let {ai , i ≥ 1} be a sequence of real numbers. We consider the following conditions.
H1 There exist some δ with 0 < δ < 1 and some α with 0 < α < 2 such that ni1 |ani |α Onδ , and assume further that EXni 0 if 1 < α < 2.
q/p
H2 There exists some p ∈ 3 · 2k−1 , 4 · 2k−1 such that ∞
i < ∞, where integer
i1 q
number k ≥ 1.
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Discrete Dynamics in Nature and Society
H3 For some h > 0 and γ > 0,
E exp h|X|γ < ∞.
2.1
H4 There exist some δ with 0 < δ < 1 and some α with 0 < α < 2 such that ni1 |ai |α Onδ , and assume further that EXn 0 if 1 < α < 2.
H5 There exists some α with 0 < α < 2 such that ni1 |ani |α On and assume further
that EXni 0 if 1 < α < 2.
H6 There exists some α with 0 < α < 2 such that ni1 |ai |α On and assume further
that EXn 0 if 1 < α < 2.
Our main results are as follows.
Theorem 2.1. Let {Xni : i ≥ 1, n ≥ 1} be an array of rowwise AANA random variables which is
stochastically dominated by a random variable X, and let {ani : i ≥ 1, n ≥ 1} be an array of real
numbers. Suppose that the conditions (H1 )–(H3 ) are satisfied. Then, for any ε > 0,
j
sα−2
>
εb
n
P
max
a
X
< ∞,
ni
ni
n
n1
i1
∞
2.2
1≤j≤n
.
where s ≥ 1/α and bn n1/α log1/γ n.
Proof. For fixed n ≥ 1, define
n
Xi
−bn IXni < −bn Xni I|Xni | ≤ bn bn IXni > bn ,
j
n
n
n
X
, j 1, 2, . . . , n.
a
−
EX
Tj ni
i
i
i1
i ≥ 1,
2.3
It is easy to check that for any ε > 0,
j
j
n max
i1 ani Xni > εbn ⊂ max|Xni | > bn
max
i1 ani Xi > εbn ,
1≤j≤n
1≤i≤n
1≤j≤n
2.4
which implies that
j
P max
i1 ani Xni > εbn
1≤j≤n
j
n ≤ P max|Xni | > bn P max
i1 ani Xi > εbn
1≤i≤n
≤
1≤j≤n
j
n n P |Xni | > bn P max
Tj > εbn − max
i1 ani EXi .
i1
n
1≤j≤n
1≤j≤n
2.5
Discrete Dynamics in Nature and Society
5
Firstly, we will show that
j
n bn−1 max
i1 ani EXi −→ 0,
1≤j≤n
By
n
i1 |ani |
α
as n −→ ∞.
2.6
Onδ and Hölder’s inequality, we have for 1 ≤ k < α that
n
k
|a | ≤
i1 ni
n
i1
|ani |
k
α/k k/α n
α−k/α
1
≤ Cn.
i1
2.7
Hence, when 1 < α < 2, we have by EXni 0, 1.6 of Lemma 1.6, 2.7 taking k 1,
Markov’s inequality, and 2.1 that
j
n
j
n −1
max
a
EX
I|X
b
>
b
>
b
bn−1 max
i1 ani EXi ≤
|X
|a
|P
|
|
ni
ni
n
ni
ni
ni
n
n
i1
i1
1≤j≤n
1≤j≤n
n
n
|ani |P |X| > bn bn−1 i1 |ani |E|Xni |I|Xni | > bn n
E exp h|X|γ
≤ Cn
γ Cbn−1 i1 |ani |E|X|I|X| > bn exp hbn
≤C
i1
Cn
Cbn−1 nE|X|I|X| > bn γ/α
nhn
∞
Cn
Cbn−1 n kn E|X|Ibk < |X| ≤ bk1 γ/α
hn
n
∞
Cn
≤
Cbn−1 n kn bk1 P |X| > bk γ/α
nhn
∞
E exp h|X|γ
Cn
−1
≤
Cbn n kn bk1
γ
γ/α
nhn
exp hbk
≤
∞
1/γ −hkγ/α
Cn
Cbn−1 n kn k 11/α logk 1
k
γ/α
hn
n
∞
1/γ −hkγ/α
Cn
≤
Cbn−1 kn k 11/α1 logk 1
k
γ/α
hn
n
≤
≤
−1/γ
Cn
Cn−1/α log n
−→ 0,
γ/α
nhn
as n −→ ∞.
2.8
Elementary Jensen’s inequality implies that for any 0 < s < t,
n
|a |t
i1 ni
1/t
≤
n
|a |s
i1 ni
1/s
.
2.9
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Discrete Dynamics in Nature and Society
Therefore, when 0 < α ≤ 1, we have by 1.5 of Lemma 1.6, 2.9, Markov’s inequality, and
2.1 that
j
n
n
n bn−1 max
i1 ani EXi ≤
|a |P |Xni | > bn bn−1 i1 |ani |E|Xni |I|Xni | ≤ bn i1 ni
1≤j≤n
≤C
n
i1
Cbn−1
|ani |P |X| > bn n
i1
|ani |E|X|I|X| ≤ bn bn P |X| > bn ≤ Cbn−1 nδ/α E|X|I|X| ≤ bn Cnδ/α P |X| > bn ≤
Cbn−1 nδ/α
≤ Cbn−1 nδ/α
≤
Cbn−1 nδ/α
≤ Cbn−1 nδ/α
n
k2
E|X|Ibk−1
Cnδ/α E exp h|X|γ
< |X| ≤ bk γ
exp hbn
n
b P |X| > bk−1 k2 k
Cnδ/α
γ/α
nhn
E exp h|X|γ
Cnδ/α
b
γ γ/α
k2 k
nhn
exp hbk−1
n
n
k2
γ/α
k1/α log k1/γ k − 1−hk−1
Cnδ/α
γ/α
nhn
−1/γ δ/α Cnδ/α
≤ Cn−1/α log n
n
γ/α
nhn
Clog n−1/γ nδ/α−1/α Cnδ/α
−→ 0,
γ/α
nhn
as n −→ ∞.
2.10
By 2.8 and 2.10, we can get 2.6 immediately. Hence, for n large enough,
j
n
n ε
P max
i1 ani Xni > εbn ≤
b
>
P
.
P
max
>
b
|X
|
T
ni
n
n
i1
1≤j≤n
1≤j≤n j
2
2.11
To prove 2.2, we only need to show that
. ∞ sα−2 n
I
n
P |Xni | > bn < ∞,
n1
i1
. ∞ sα−2
n ε
n
P max
Tj > bn < ∞.
J
n1
1≤j≤n
2
2.12
Discrete Dynamics in Nature and Society
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By Definition 1.3, Markov’s inequality and 2.1, we can see that
. ∞ sα−2 n
I
n
P |Xni | > bn n1
i1
≤C
≤C
≤C
∞
nsα−2
n1
∞
n1
sα−1
n
n
i1
P |X| > bn E exp h|X|γ
γ
exp hbn
∞ nsα−1
< ∞.
n1 hnγ/α
n
2.13
n
For fixed n ≥ 1, it is easily seen that {Xi , 1 ≤ i ≤ n} are still AANA random variables by
Lemma 1.4. For r > 2, it follows from Lemma 1.5, Cr ’s inequality, and Jensen’s inequality that
. ∞ sα−2
n ε
J
n
P max
Tj > bn
n1
1≤j≤n
2
∞
n r
sα−2 −r
bn E max
Tj ≤ C n2 n
1≤j≤n
≤C
≤C
∞
nsα−2 bn−r
n2
∞
nsα−2 bn−r
n2
n
r/2
n
n r
n 2
2 n
E
−
EX
E
−
EX
|a | Xi
|a | Xi
i i i1 ni
i1 ni
n
r
2 r/2
∞
n
n r
2 n sα−2 −r
bn
|a | E
Xi C n2 n
|a | E
Xi i1 ni
i1 ni
n
r
.
J1 J2 .
2.14
Taking r > max{2, αsα − 1/1 − δ}, which implies that r > α. It follows from Cr ’s inequality,
1.5 of Lemma 1.6, 2.9, Markov’s inequality, and 2.1 that
n
. ∞
n r
J1 C n2 nsα−2 bn−r i1 |ani |r E
Xi ≤C
∞
nsα−2 bn−r
n2
∞
n
i1
|ani |r E|Xni |r I|Xni | ≤ bn bnr P |Xni | > bn n
|ani |r E|X|r I|X| ≤ bn bnr P |X| > bn ∞
∞
≤ C n2 nsα−2rδ/α bn−r E|X|r I|X| ≤ bn C n2 nsα−2rδ/α P |X| > bn ≤C
≤C
≤C
nsα−2 bn−r
n2
∞
i1
nsα−2rδ/α bn−r
n2
∞ ∞
k2
n
k2
γ
E|X| Ibk−1 < |X| ≤ bk C
∞
n2
sα−2rδ/α
n
E exp h|X|γ
γ
exp hbn
∞ nsα−2rδ/α
−r/γ r
sα−2rδ/α −r/α
n
n
b
P
>
b
C
log
n
|X|
k−1
γ/α
k
nk
n2
nhn
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Discrete Dynamics in Nature and Society
≤C
≤C
∞
br
k2 k
∞ nsα−2rδ/α
E exp h|X|γ
γ C n2
γ/α
nhn
exp hbk−1
∞ kr/α log k r/γ
k2
γ/α
k − 1hk−1
C
∞ nsα−2rδ/α
< ∞.
γ/α
n2
nhn
2.15
By Cr ’s inequality, 1.5 of Lemma 1.6, 2.9, and Jensen’s inequality, we can get that
2 r/2
n
. ∞
2 n J2 C n2 nsα−2 bn−r
|a
|
E
Xi i1 ni
≤C
≤C
≤C
≤C
≤C
r/2
2
2
2
E|X
|a
|
I|X
P
b
≤
b
>
b
|X
|
|
|
ni
ni
ni
n
ni
n
n
i1
∞
n
∞
n
nsα−2 bn−r
n2
nsα−2 bn−r
n2
r/2
2
2
2
EX
I|X|
≤
b
b
P
>
b
|X|
|a
|
ni
n
n
n
i1
r/2
sα−2rδ/α −r
2
2
EX
n
b
I|X|
≤
b
P
>
b
b
|X|
n
n
n
n
n2
∞
r/2
∞
sα−2rδ/α −r
2
EX
n
b
I|X|
≤
b
C
nsα−2rδ/α P |X| > bn r/2
n
n
n2
n2
∞
∞
<∞
nsα−2rδ/α bn−r E|X|r I|X| ≤ bn C
n2
∞
n2
nsα−2rδ/α P |X| > bn see the proof of 2.15 .
2.16
Therefore, the desired result 2.2 follows from 2.13–2.16 immediately. This completes the
proof of the theorem.
Similar to the proof of Theorem 2.1, we can get the following result for sequences of
AANA random variables.
Theorem 2.2. Let {Xn , n ≥ 1} be a sequence of AANA random variables which is stochastically
dominated by a random variable X, and let {ani , i ≥ 1, n ≥ 1} be an array of real numbers. Suppose
that the conditions (H1 )–(H3 ) are satisfied (EXni 0 is replaced by EXn 0 in H1 ). Then, for any
ε > 0,
j
sα−2
>
εb
< ∞,
n
P
max
a
X
i1 ni i n
n1
∞
1≤j≤n
2.17
.
where s ≥ 1/α and bn n1/α log1/γ n.
The following result provides the Marcinkiewicz-Zygmund type strong law of large
numbers for weighted sums ni1 ai Xi of AANA sequence of random variables.
Discrete Dynamics in Nature and Society
9
Theorem 2.3. Let {Xn , n ≥ 1} be a sequence of AANA random variables which is stochastically
dominated by a random variable X and {an , n ≥ 1} be a sequence of real numbers. Suppose that the
conditions (H2 )–(H4 ) are satisfied. Then for any ε > 0,
∞
n1
sα−2
n
P max
Sj > εbn
1≤j≤n
Sn
lim
n → ∞ bn
< ∞,
2.18
2.19
0 a.s.,
.
where s ≥ 1/α, bn n1/α log1/γ n and Sn ni1 ai Xi for n ≥ 1.
Proof. Similar to the proof of Theorem 2.1, we can get 2.18 immediately, which yields that
−1
Sj > εbn < ∞.
n
P
max
n1
∞
2.20
1≤j≤n
Therefore,
∞>
−1
Sj > εbn
n
P
max
n1
∞
1≤j≤n
∞ 2i1 −1
i0
n2i
1 ∞
≥
P
2 i1
1/γ
n−1 P max
Sj > εn1/α log n
1≤j≤n
1/γ
max Sj > ε2i1/α log 2i1
1≤j≤2i
2.21
.
By Borel-Cantelli lemma, we obtain that
max1≤j≤2i Sj lim
1/γ 0 a.s.
i → ∞ i1/α
2
log 2i1
2.22
For all positive integers n, there exists a positive integer i0 such that 2i0 −1 ≤ n < 2i0 . We have
by 2.22 that
22/α max1≤j≤2i Sj i0 1 1/γ
|Sn |
|Sn |
≤ max
≤
−→ 0 a.s., as i0 −→ ∞,
1/γ i − 1
bn
2i0 −1 ≤n<2i0 bn
0
2i0 1/α log 2i0 1
2.23
which implies 2.19. This completes the proof of the theorem.
Remark 2.4. In Theorems 2.1–2.3, the condition H1 or H4 is needed. Under the weaker
condition H5 or H6 than H1 or H4 , we can get the following Theorems 2.5–2.7. The
details of their proofs are omitted.
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Discrete Dynamics in Nature and Society
Theorem 2.5. Let {Xni : i ≥ 1, n ≥ 1} be an array of rowwise AANA random variables which is
stochastically dominated by a random variable X, and let {ani : i ≥ 1, n ≥ 1} be an array of real
numbers. Suppose that the conditions (H2 ), (H3 ), and (H5 ) are satisfied. Then, for any ε > 0,
j
−1
>
εb
n
P
max
a
X
< ∞,
n
n1
i1 ni ni
∞
1≤j≤n
2.24
.
where bn n1/α log1/γ n.
Theorem 2.6. Let {Xn , n ≥ 1} be a sequence of AANA random variables which is stochastically
dominated by a random variable X, and let {ani , i ≥ 1, n ≥ 1} be an array of real numbers. Suppose
that the conditions (H2 ), (H3 ) and (H5 ) are satisfied (EXni 0 is replaced by EXn 0 in (H5 )). Then,
for any ε > 0,
j
−1
>
εb
< ∞,
n
P
max
a
X
i1 ni i n
n1
∞
1≤j≤n
2.25
.
where bn n1/α log1/γ n.
Theorem 2.7. Let {Xn , n ≥ 1} be a sequence of AANA random variables which is stochastically
dominated by a random variable X, and let {an , n ≥ 1} be a sequence of real numbers. Suppose that
the conditions (H2 ), (H3 ), and (H6 ) are satisfied. Then, for any ε > 0,
∞
n P max
Sj > εbn
n1
−1
1≤j≤n
Sn
lim
0 a.s.,
n → ∞ bn
< ∞,
2.26
.
where bn n1/α log1/γ n and Sn ni1 ai Xi for n ≥ 1.
Acknowledgments
The authors are most grateful to the Editor Wei-Der Chang and anonymous referees for
careful reading of the paper and valuable suggestions which helped in improving an
earlier version of this paper. This work was supported by the NNSF of China 11171001,
Provincial Natural Science Research Project of Anhui Colleges KJ2010A005, Talents Youth
Fund of Anhui Province Universities 2010SQRL016ZD, Youth Science Research Fund of
Anhui University 2009QN011A, and the academic innovation team of Anhui University
KJTD001B.
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Discrete Dynamics in Nature and Society
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