RESEARC H Open Access
A note on the complete convergence for
sequences of pairwise NQD random variables
Haiwu Huang
1,2*
, Dingcheng Wang
1
, Qunying Wu
2
and Qingxia Zhang
1
* Correspondence:
[email protected]
1
School of Mathematics Science,
University of Electronic Science
and Technology of China,
Chengdu 610054, PR China
Full list of author information is
available at the end of the article
Abstract
In this paper, complete convergence and strong law of large numbers for sequences
of pairwise negatively quadrant dependent (NQD) random variables with non-
identically distributed are investigated. The results obtained generalize and extend
the relevant result of Wu (Acta. Math. Sinica. 45(3), 617-624, 2002) for sequences of
pairwise NQD random variables with identically distributed.
2000 MSC: 60F15.
Keywords: pairwise NQD random variable sequences, complete convergence, almost
sure convergence
1 Introduction
In many st ochastic models, the assumption of independence among random variables

dom variables and negatively associated (NA) [2] random variables are the most
important and special cases of pairwise NQD random variables. So, it is very significant
to study probabilistic properties of this wider pairwise NQD class. Since the concept of
NQD random variables was introduced by Lehmann, many r esearchers have been
established a large number of limit results for pairwise NQD random variable
sequences. We can refer to Matula [3] for the Kolmogo rov strong law of large num-
bers, Wang et al. [4] for the Marcinkiewicz strong law of large numbers and Baum and
Katz complete convergence theorem, Wu [5] for Three series theorem, the complete
convergence theorem and Marcinkiewicz strong law of large numbers, Chen [6] for the
Huang et al. Journal of Inequalities and Applications 2011, 2011:92
http://www.journalofinequalitiesandapplications.com/content/2011/1/92
© 201 1 Huang et al; licensee Springer. This is an Op en Access article distributed under the terms of the Creative Commons Attribution
License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly ci ted.
Kolmogorov-Chung-type strong law of large numbers, Gan and Chen [7] for the strong
stability of Jamison’s weig hted sums. But most of th eir results were achieved under the
identically distributed condition and some results were obtained even under the condi-
tion of * (1) <1, where
ϕ
∗
(1) = lim
m→∞
sup
n≥m
sup
A∈F
n
m
,B∈F
∞

Theorem A [10] Let ap ≥ 1, p>2, and let {X
n
; n ≥ 1} be a sequence of independent
and identically distributed random variables and E|X
1
|
p
< ∞.If
1
2
<α≤ 1
,assumethat
EX
n
=0,n ≥ 1. Then
∞

n=1
n
αp−2
P(max
1≤j≤n






j


n
=0,n ≥ 1. Then
∞

n=1
n
αp−2
P
⎛
⎝
max
1≤j≤n






j

i=1
X
i






>εn

n
).
We will use the following concept in this article. Let {X
n
; n ≥ 1}beasequenceof
NQD random variables and let X be a n onnegative random variable. If there exists a
constant such that
sup
n
≥
1
P( |X
n
|≥t) ≤ cP(X ≥ t)forallt ≥ 0
.
Then, {X
n
; n ≥ 1} is said to be stochastically dominated by X (briefly {X
n
; n ≥ 1} π X).
Clearly if {X
n
; n ≥ 1} π X,thenfor0<p < ∞, E |X
n
|
p
≤ cEX
p
for any n ≥ 1. Now we
state the main results of this article.

|S
j
| >εn
1/p
log n

< ∞ for all ε>0
.
(2:1)
Theorem 2.2 Let {X
n
; n ≥ 1} be a sequence of non-identically distributed pairwise
NQD random variables with {X
n
; n ≥ 1} π X and E|X|
p
< ∞ ,0<p <2andlet
S
n
=

n
i
=1
X
. When 1 ≤ p < 2, assumes that EX
n
=0,n ≥ 1. Then
lim
n→∞

E

n

i=1
X
i

2
≤
n

i=1
EX
2
i
, for all n ≥ 1;
(2)
E max
1≤j≤n

j

i=1
X
i

2
≤
4log

) − n
1/p
I(X
i
< −n
1/p
)
,
S
(n)
j
=
j

i
=1
X
(n)
i
, for any i ≥ 1. For all ε >0, first we show that
n
−1/p
max
1≤j≤n





j

EX
(n)
i






≤ n
−1/p
n

i=1



EX
(n)
i



= n
−1/p
n

i=1



1/p
)+2n
E|X|
n
1/p
≤ n
1−1/p
E|X|I(|X|≤n
1/p
)
= n
1−1/p
n

k
=1
E|X|I(k − 1 < |X|
p
≤ k)
(3:2)
Since,
∞

k=1
k
1−1/p
E|X|I(k − 1 < |X|
p
≤ k)=
∞

I(k − 1 < |X|
p
≤ k)
= E
|
X
|
p
< ∞.
(3:3)
It follows from Kronecker lemma that
n
1−1/p
n

k
=1
E|X|I(k − 1 < |X|
p
≤ k) → 0asn →∞
.
(3:4)
Hence, we get that
n
−1/p
max
1≤j≤n







j

i=1
EX
(n)
i






≤ n
−1/p
n

i=1



EX
(n)
i



= n

E(|X|I(|X| > n
1/p
)+n
1/p
I(|X| > n
1/p
))
 n
1−1/p
E|X|
p
n
(1−p)/p
I(|X| > n
1/p
)
= E|X|
p
I
(
|X| > n
1/p
)
→ 0
(3:6)
From (3.5) and (3.6), we easily know that (3.1) follows.
Huang et al. Journal of Inequalities and Applications 2011, 2011:92
http://www.journalofinequalitiesandapplications.com/content/2011/1/92
Page 4 of 8
By (3.1), for any ε >0, n large enough, it follows from that

log n,





n

i=1
EX
(n)
i





<
ε
2
n
1/
p
(3:7)
It follows from (3.1) that for n large enough
P

max
1≤j≤n
|S

>
ε
2
n
1/p
log n

(3:8)
Hence, we need only to prove that
I 
∞

n=1
n
−1
n

j
=1
P( |X
j
| > n
1/p
) < ∞
.
(3:9)
and
II 
∞


n=1
n
−1
n

j
=1
P( |X
j
| > n
1/p
) <
∞

n=1
P( |X| > n
1/p
)  E|X|
p
< ∞
.
(3:11)
Denote that
˜
X
(n)
i
= X
(n)
i

=
0
.It
follows from Lemma 3.2 and Markov inequality that
II 
∞

n=1
n
−1
P

max
1≤j≤n



S
(n)
j
− ES
(n)
j



>
ε
2
n

n=1
n
−1−2α
log
−2
nlog
2
n
n

i=1
E(
˜
X
(n)
i
)
2
≤ c
∞

n=1
n
−2α
E|X|
2
I(|X |≤n
1/p
)+c
∞


n=k
n
−2α
≤ c
∞

k=1
k
−2α+1
E|X|
2
I(k − 1 < |X|
p
≤ k)
= c
∞

k=1
k
−2α+1
E|X|
p
|X|
2−p
I(k − 1 < |X|
p
≤ k)
≤ c
∞

The proof of Theorem 2.1 is complete.
Remark 3.1 Theorem 2.1 shows that when ap = 1, Baum and Katz complete conver-
gence theorem for pairwise NQD random variable sequences still holds true under the
strong er condition. The result generalizes and extends the corr esponding result of Wu
[5] for sequences of pairwise NQD random variables with identically distributed.
Remark 3 .2 Under the conditi ons of Theorem 2.1, it is well known that (2.1) holds
for 0 <p < 1 without the factor log n, i.e.
∞

n
=1
n
−1
P(max
1≤j≤n
|S
j
| >εn
1/p
) < ∞ for 0 < p < 1 and any ε>0
.
Proof of Theorem 2.2 For all ε >0, from (2.1), we obtain that
∞

n
=1
n
−1
P(max
1≤j≤n

i+1
− 1)
−1
P(max
1≤j≤n
|S
j
| >εn
1/p
log n
)
≥
1
2
∞

i
=1
P(max
1≤j≤2
i
|S
j
| >ε2
(i+1)/p
log(2
i+1
)).
(3:13)
It follows from Borel-Cantelli lemma that

i+1
)
=0. a.s
.
(3:15)
For all positive integers n, t here exists a non-negative integer i
0
, such that
2
i
0
−1
≤
n < 2
i
0
.
Thus
max
2
i
0
−1
≤n≤2
i
0
|S
n
|
n

(i
0
+1)/p
log(2
i
0
+1
)

i
0
+1
i
0
− 1

→
0. a.s.
(3:16)
We have
lim
n→∞
|
S
n
|
n
1/p
lo
g


n
=1
n
−1
P(max
1≤j≤n
|S
j
| >εn
1/p
log n) < ∞ for all ε>0
.
Corollary 3.6 Let {X
n
; n ≥ 1} be a sequence of non-identically distributed NOD (or
NA) random variables satisfying the conditions of Corollary 3.3, then,
lim
n→∞
S
n
n
1/p
lo
g
n
=0. a.s
.
Acknowledgements
This study was supported by the National Natural Science Foundation of China (11061012), Project supported by

doi:10.1073/pnas.33.2.25
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http://www.journalofinequalitiesandapplications.com/content/2011/1/92
Page 7 of 8
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doi:10.1090/S0002-9947-1965-0198524-1
doi:10.1186/1029-242X-2011-92
Cite this article as: Huang et al.: A note on the complete convergence for sequences of pairwise NQD random
variables. Journal of Inequalities and Applications 2011 2011:92.
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