Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2010, Article ID 218380, 15 pages
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, hushuhe@263.net
Received 20 July 2010; Revised 9 October 2010; Accepted 2 November 2010
Academic Editor: Ibrahim Yalcinkaya
Copyright q 2010 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 variables X1 , X2 , . . . , Xn is said to be negatively
associated NA 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 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 Proschan 1 and
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 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 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
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Discrete Dynamics in Nature and Society
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 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. 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, and Wang et al. 7
established the law of the iterated logarithm for product sums. Recently, Yuan and An 8
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 Sn ni1 Xi . Let X 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 an Obn denotes that there exists a
positive constant C such that |an /bn | ≤ 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 let f1 , f2 , . . . be all nondecreasing (or
nonincreasing) functions, then {fn Xn , 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 EXn 0 for each n ≥ 1. If
∞ 2
< ∞, then there exists a positive constant Cp depending only on p such
n1 q n
that
n
E max|Si |p ≤ Cp E|Xi |p
1≤i≤n
for all n ≥ 1, where Cp 2p 22−p p 6pp i1
∞
n1
p/q
q2 n
.
1.3
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3
Proof. We use the same notations as that in the study by Yuan and An 8. They proved that
p
p
n
n−1
p p
2−p
2/q
Emax Si ≤ 2 p Xi p 6p
q iXi p ,
1≤i≤n
i1
i1
p
p
n
n−1
p p
2−p
2/q
q iXi p ,
Emax−Si ≤ 2 p Xi p 6p
1≤i≤n
i1
1.4
i1
p
p
max|Si |p ≤ 2p−1 max Si 2p−1 max−Si .
1≤i≤n
1≤i≤n
1≤i≤n
By 1.4 and Hölder’s inequality, we have
p
p
E max|Si |p ≤ 2p−1 Emax Si 2p−1 Emax−Si 1≤i≤n
1≤i≤n
≤2
p
2−p
2
⎡
1≤i≤n
p n
n
p p
2/q
p E|Xi | 6p
q iXi p
i1
i1
⎤
p/q
n
n
n
p
≤ 2p ⎣22−p p E|Xi |p 6p
q2 i
E|Xi |p ⎦
i1
i1
1.5
i1
⎡
p/q ⎤
∞
n
n
p
⎦ E|Xi |p Cp E|Xi |p .
≤ 2p ⎣22−p p 6p
q2 n
n1
i1
i1
This completes the proof of the lemma.
We point out that Lemma 1.4 has been studied by Yuan and An 8. But here we give
the accurate coefficient Cp . And Lemma 1.4 generalizes and improves the result of Lemma
2.2 in the study by Ko et al. 6.
Lemma 1.5 cf. Fazekas and Klesov 9, Theorem 2.1 and Hu et al. 10, Lemma 1.5. Let
{Xn , n ≥ 1} be a sequence of random variables. Let b1 , b2 , . . . be a nondecreasing unbounded sequence
of positive numbers, and let α1 , α2 , . . . be nonnegative numbers. Let r and C be fixed positive numbers.
Assume that, for each n ≥ 1,
n
E max|Sl |r ≤ C αl ,
1≤l≤n
∞
αl
l1
1.6
l1
blr
< ∞,
1.7
then
lim
Sn
n → ∞ bn
0 a.s.,
1.8
4
Discrete Dynamics in Nature and Society
and with the growth rate
βn
Sn
a.s.,
O
bn
bn
1.9
where
βn max bk vkδ/r ,
∀0 < δ < 1,
1≤k≤n
vn ∞
αk
br
kn k
,
βn
0,
n → ∞ bn
lim
r n
Sl αl
E max ≤ 4C
r < ∞,
b
1≤l≤n bl
l1 l
r ∞
Sl αl
≤ 4C
E sup r < ∞.
b
b
l
l≥1
l1 l
1.10
If further one assumes that αn > 0 for infinitely many n, then
r ∞
Sl αl
< ∞.
E sup ≤ 4C
βr
l≥1 βl
l1 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. Denote Λk α1 α2 · · · αk for k ≥ 1. Let r be a fixed positive number
satisfying 1.6. If
∞
Λl
l1
1
1
− r
blr bl1
< ∞,
Λn
is bounded,
bnr
1.12
1.13
then 1.8–1.11 hold.
Lemma 1.7 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,
E max|Si |p
1≤i≤n
⎧
n
⎨
p/2 ⎫
n
⎬
E|Xi |p EXi2
.
≤ Dp
⎩ i1
⎭
i1
1.14
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5
Lemma 1.8. Assume that the inverse function ψx of φx satisfies
ψn
If Eφ|X| < ∞, then
n
1
On.
ψi
i1
∞
n1 1/ψnE|X|I|X|
1.15
> ψn < ∞.
Proof. Since ψx is an increasing function of x, we have that
∞
∞
∞
1
1 E|X|I |X| > ψn E|X|I ψi < |X| ≤ ψi 1
ψn
ψn
in
n1
n1
∞
i
1
E|X|I ψi < |X| ≤ ψi 1
ψn
i1
n1
≤
∞
i
1
P ψi < |X| ≤ ψi 1 ψi 1
ψn
i1
n1
≤C
1.16
∞
P ψi < |X| ≤ ψi 1 i
i1
≤ CE φ|X| < ∞.
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
∞ 2
n1 q n < ∞, and let {bn , n ≥ 1} be a nondecreasing unbounded sequence of positive numbers;
1 < p ≤ 2. Assume that
∞
E|Xn |p
p
n1
bn
< ∞,
2.1
then
lim
Sn
n → ∞ bn
0 a.s.,
2.2
and with the growth rate
βn
Sn
O
a.s.,
bn
bn
2.3
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Discrete Dynamics in Nature and Society
where
βn max bk vkδ/2 ,
∀0 < δ < 1,
1≤k≤n
∞
αk
vn p
kn bk
,
lim
βn
n → ∞ bn
0,
αk Cp E|Xk |p , k ≥ 1, Cp is defined in Lemma 1.4,
p n
Sl αl
E max ≤ 4
p < ∞,
1≤l≤n bl
b
l1 l
p ∞
Sl αl
≤4
E sup p < ∞.
b
l
l≥1
l1 bl
2.4
If further one assumes that αn > 0 for infinitely many n, then
p ∞
Sl αl
≤4
E sup p < ∞.
l≥1 βl
l1 β
2.5
l
Proof. By Lemma 1.4, we have
n
n
E max |Sk |p ≤ Cp E|Xk |p αk .
1≤k≤n
k1
2.6
k1
It follows from 2.1 that
∞
αn
p
n1 bn
Cp
∞
E|Xn |p
p
n1
bn
< ∞.
2.7
Thus, 2.2–2.5 follow from 2.6, 2.7, and Lemma 1.5 immediately. We complete the proof
of the theorem.
Theorem 2.2. Let {Xn , n ≥ 1} be a sequence of AANA random variables with
1 ≤ p < 2. Denote Qn max1≤k≤n EXk2 for n ≥ 1 and Q0 0. Assume that
∞
Qn
< ∞,
2/p
n1 n
∞
n1
q2 n < ∞,
2.8
then
n
1 Xi − EXi 0
n → ∞ n1/p
i1
lim
a.s.,
2.9
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7
and with the growth rate
n
βn
1 −
EX
O
X
i
i
n1/p i1
n1/p
2.10
a.s.,
where
βn max k1/p vkδ/2 ,
1≤k≤n
∀0 < δ < 1,
vn ∞
αk
,
2/p
k
kn
βn
0,
n → ∞ n1/p
lim
αk C2 kQk − k − 1Qk−1 , k ≥ 1, C2 is defined in Lemma 1.4,
n
S l 2
αl
E max 1/p < ∞,
≤4
2/p
1≤l≤n l
l1 l
∞
S l 2
αl
≤4
E sup 1/p < ∞.
2/p
l
l
l≥1
l1
2.11
2.12
2.13
If further one assumes that αn > 0 for infinitely many n, then
2 ∞
Sl αl
≤4
< ∞.
E sup 2
β
l
l≥1
l1 βl
2.14
∞
S l r
4r αl
≤1
E sup 1/p < ∞.
2/p
2 − r l1 l
l≥1 l
2.15
In addition, for any r ∈ 0, 2,
Proof. Assume that EXn 0, bn n1/p , and Λn that
n
l1
αl , n ≥ 1. By Lemma 1.4, we can see
2 ⎞
k
n
n
αk .
E⎝ max Xi ⎠ ≤ C2 EXi2 ≤ C2 nQn 1≤k≤n
i1
i1
k1
⎛
2.16
It is a simple fact that αk ≥ 0 for all k ≥ 1. It follows from 2.8 that
∞
l1
Λl
1
1
− 2
2
bl
bl1
C2
∞
l1
lQl
1
l2/p
−
1
l 12/p
≤
∞
2C2 Ql
< ∞.
p l1 l2/p
2.17
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Discrete Dynamics in Nature and Society
That is to say that 1.12 holds. By Remark 2.1 in Fazekas and Klesov 9, 1.12 implies 1.13.
By Lemma 1.6, we can obtain 2.9–2.14 immediately. By 2.13, it follows that
∞ ∞ S l r
S l r
Sl 1/r
dt
P sup 1/p > t dt ≤ 1 P sup 1/p > t
E sup 1/p l
l
l
0
l≥1
1
l≥1
l≥1
∞
∞
S l 2
αl
4r ≤ 1 E sup 1/p t−2/r dt ≤ 1 < ∞.
2/p
2
−
r
l
l
1
l≥1
l1
2.18
The proof is complete.
Theorem 2.3. Let p ∈ 3 · 2k−1 , 4 · 2k−1 , where integer number k ≥ 1, and let {Xn , n ≥ 1} be
q/p
n < ∞. Let
a sequence of AANA random variables with EXi 0 for all i ≥ 1 and ∞
n1 q
{bn , n ≥ 1} be a nondecreasing unbounded sequence of positive numbers. Assume that
∞ p/2−1
n
p
bn
n1
∞
E|Xk |
∞
np/2−2
p
k1
E|Xn |p < ∞,
p
bn
nk1
2.19
< ∞,
then 1.8–1.11 hold (for C 1), where
⎛
α1 2Dp E|X1 |p ,
⎞
k−1 k
p
p
αk 2Dp ⎝kp/2−1 EXj − k − 1p/2−1 EXj ⎠,
j1
k ≥ 2,
2.20
j1
and Dp is defined in Lemma 1.7.
Proof. Since p > 2, 0 < 2/p < 1. By Cr ’s inequality,
n
2/p
|Xi |
p
≤
i1
n
Xi2 ,
2.21
i1
which implies that
n
i1
n
E|Xi | ≤ E
Xi2
p
p/2
.
2.22
i1
By Jensen’s inequality, we have
p/2
p/2
n
n
2
2
EXi
≤E
Xi
.
i1
i1
2.23
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9
By 2.22-2.23 and Cr ’s inequality,
n
E|Xi |p i1
n
p/2
EXi2
p/2
n
n
2
≤ 2E
Xi
≤ 2np/2−1 E|Xi |p .
i1
i1
2.24
i1
It follows from Lemma 1.7 and 2.24 that
n
n
αl .
E max |Si |p ≤ 2Dp np/2−1 E|Xi |p 1≤i≤n
i1
2.25
l1
It is a simple fact that
⎞
⎛
k−1 p
0 ≤ αk ≤ C1 p ⎝kp/2−1 E|Xk |p kp/2−2 EXj ⎠,
2.26
j1
where C1 p is a positive number depending only on p and Dp . By 2.19,
∞
∞ p/2−1
∞
n
np/2−2
p
p
E|Xk |
< ∞.
p ≤ C1 p
p E|Xn | p
n1 bn
n1 bn
k1
nk1 bn
∞
αn
2.27
The desired results follow from 2.25–2.27 and Lemma 1.5 immediately.
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
2
2
with ∞
n1 q n < ∞, EX 0, EX < ∞, and Eφ|X| < ∞. Assume that the inverse function
ψx of φx satisfies 1.15. Let {ani , n ≥ 1, i ≥ 1} be a triangular array of positive constants such
that
i max1≤i≤n ani O1/ψn,
ii ni1 a2ni Olog−1−α n for some α > 0.
Then, for any ε > 0,
∞
n1
−1
n P
j
max ani Xi > ε < ∞.
1≤j≤n
i1
3.1
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Proof. For each n ≥ 1, denote
n
Xj
−ψnI Xj < −ψn Xj I Xj ≤ ψn ψnI Xj > ψn ,
n
j
Tj
n
ani Xi
n
− Eani Xi
!
1 ≤ j ≤ n,
1 ≤ j ≤ n,
,
i1
A
n
"
n
Xi Xi
!
i1
n
"
|Xi | ≤ ψn ,
i1
BA
n
#
n
Xi /
Xi
!
i1
j
max ani Xi > ε .
1≤j≤n
i1
n
#
|Xi | > ψn ,
i1
En 3.2
It is easy to check that
j
n
ani Xi Tj
i1
j
j
n
Eani Xi ani Xi I |Xi | > ψn
i1
j
i1
ani ψn I Xj < −ψn − I Xj > ψn ,
i1
j
n n maxT Eani Xi > ε En B
1≤j≤n j
i1
En En A En B 3.3
j
n n maxTj > ε − max Eani Xi B.
1≤j≤n
1≤j≤n
i1
⊂
Therefore,
j
n n maxTj > ε − max Eani Xi P B
1≤j≤n
1≤j≤n
i1
P En ≤ P
j
n
n n maxTj > ε − max Eani Xi P |Xi | > ψn .
1≤j≤n
1≤j≤n
i1
i1
≤P
3.4
Firstly, we will show that
j
n max Eani Xi −→ 0,
1≤j≤n
i1
as n −→ ∞.
3.5
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11
It follows from Lemma 1.8 and Kronecker’s lemma that
n
1 E|X|I |X| > ψi −→ 0,
ψn i1
as n −→ ∞.
3.6
By EX 0, condition i, 3.6, and ψn ↑ ∞, we can see that
j
n n
n Eani Xi I |Xi | ≤ ψn ani ψnEI |Xi | > ψn
max Eani Xi ≤
i1
1≤j≤n
i1
i1
≤
n
n
ani E|Xi |I |Xi | > ψn ani E|Xi |I |Xi | > ψn
i1
≤
3.7
i1
n
C E|X|I |X| > ψi −→ 0,
ψn i1
as n −→ ∞,
which implies 3.5. By 3.4 and 3.5, we can see that, for sufficiently large n,
j
n
n ε
max ani Xi > ε ≤ P maxTj >
P |Xi | > ψn .
1≤j≤n
1≤j≤n
2
i1
i1
P
3.8
To prove 3.1, it suffices to show that
n ε
n P maxTj >
< ∞,
1≤j≤n
2
n1
∞
−1
∞
n
n−1 P |Xi | > ψn < ∞.
n1
3.9
i1
By Markov’s inequality, Lemma 1.4, Cr inequality, EX 2 < ∞, and condition ii, we have
n ε
n P maxTj >
1≤j≤n
2
n1
∞
−1
≤C
∞
∞
n
n 2
n 2
n−1 E maxTj ≤ C n−1 Eani Xi 1≤j≤n
n1
≤C
≤C
≤C
∞
n
n1
i1
n−1
n1
i1
∞
n
a2ni EX 2 I |X| ≤ ψn C n−1 a2ni ψ 2 nEI |X| > ψn
n1
i1
∞
∞
n
n
n−1 a2ni EX 2 I |X| ≤ ψn C n−1 a2ni EX 2 I |X| > ψn
n1
i1
∞
n
∞
n1
i1
n1
n−1
n1
a2ni ≤ C
n−1 log−1−α n < ∞.
i1
3.10
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Discrete Dynamics in Nature and Society
It follows from Eφ|X| < ∞ that
∞
∞
∞
n
n−1 P |Xi | > ψn P |X| > ψn P φ|X| > n
n1
i1
n1
3.11
n1
≤ CE φ|X| < ∞.
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 some p ∈ 3 · 2k−1 , 4 · 2k−1 , where integer
number k ≥ 1, 0 < t < 2, and for any ε > 0, the following conditions are satisfied:
bn
∞
!
cn P |ani Xi | ≥ εbn1/t < ∞,
n1
∞
bn
−p/t c n bn
n1
∞
∞
n1
!
|ani |p E|Xi |p I |ani Xi | < εbn1/t < ∞,
3.12
i1
−p/t
c n bn
n1
and
i1
b
! p/2
n
2
2
1/t
ani EXi I |ani Xi | < εbn
< ∞,
i1
qq/p n < ∞, then
⎧
⎫
∞
⎨
⎬
!%
i $
1/t
1/t
≥ εbn
< ∞.
anj Xj − anj EXj I anj Xj < εbn
cn P max ⎩1≤i≤bn j1
⎭
n1
3.13
Proof. Note that if the series ∞
holds. Therefore, we will
n1 cn is convergent, then 3.13
consider only such sequences {cn , n ≥ 1} for which the series ∞
n1 cn is divergent.
Let
n
Yi
!
!
!
−εbn1/t I ani Xi < −εbn1/t ani Xi I |ani Xi | < εbn1/t εbn1/t I ani Xi > εbn1/t ,
Sni i
n
Yj ,
3.14
n ≥ 1, i ≥ 1.
j1
Note that
⎧
⎨
⎫
⎬
!%
i $
P max anj Xnj − anj EXj I anj Xj < εbn1/t ≥ εbn1/t
⎩1≤i≤bn j1
⎭
≤C
bn
!
−p/t
P |ani Xi | ≥ εbn1/t 2p ε−p bn
i1
E max |Sni − ESni |
1≤i≤bn
3.15
p
.
Discrete Dynamics in Nature and Society
13
n
Using the Cr inequality and Jensen’s inequality, we can estimate E|Yi
following way:
n p
− EYi | in the
!
!
p/t
n
n p
EYi − EYi ≤ C|ani |p E|Xi |p I |ani Xi | < εbn1/t Cbn P |ani Xi | ≥ εbn1/t .
3.16
By 3.15, 3.16, and Lemma 1.7, we can get
⎧
⎨
⎫
⎬
!%
i $
P max anj Xj − anj EXj I anj Xj < εbn1/t ≥ εbn1/t
⎩1≤i≤bn j1
⎭
≤C
bn
bn
!
!
−p/t P |ani Xi | ≥ εbn1/t Cbn
|ani |p E|Xi |p I |ani Xi | < εbn1/t
i1
3.17
i1
−p/t
Cbn
b
n
p/2
a2ni EXi2 I
|ani Xi | <
εbn1/t
!
.
i1
Therefore, we can conclude that 3.13 holds by 3.12 and 3.17.
Theorem 3.3. Let 1 ≤ r ≤ 2 and let {Xn , n ≥ 1} be a sequence of AANA random variables with
EXn 0 and E|Xn |r < ∞ for n ≥ 1. Let {ani , n ≥ 1, i ≥ 1} be an array of real numbers satisfying the
condition
n
!
|ani |r E|Xi |r O nδ
as n −→ ∞
3.18
i1
q/p
and ∞
n < ∞ for some 0 < δ ≤ 2/p and p ∈ 3 · 2k−1 , 4 · 2k−1 , where integer number k ≥ 1.
n1 q
Then, for any ε > 0 and αr ≥ 1,
⎞
i
nαr−2 P ⎝max anj Xj ≥ εnα ⎠ < ∞.
1≤i≤n n1
j1
∞
⎛
3.19
14
Discrete Dynamics in Nature and Society
Proof. Take cn nαr−2 , bn n, and 1/t α in Theorem 3.2. By 3.18, we have
bn
∞
∞
∞
n
!
|ani |r E|Xi |r
cn P |ani Xi | ≥ εbn1/t ≤ C nαr−2
≤
C
n−2δ < ∞,
αr
n
n1
i1
n1
i1
n1
∞
∞
∞
n
! n−2 |ani |r E|Xi |r ≤ C n−2δ < ∞,
|ani |p E|Xi |p I |ani Xi | < εbn1/t ≤
bn
−p/t c n bn
n1
i1
n1
i1
n1
p/2
b
∞
∞
n
! p/2
n
−p/t 2
r
r
2
1/t
αr−2−αrp/2
c n bn
ani EXi I |ani Xi | < εbn
≤C n
|ani | E|Xi |
n1
i1
n1
≤C
∞
i1
αr−2−αrp/2δp/2
n
n1
≤C
∞
nαr1−p/2−1 < ∞
n1
3.20
following from δp/2 ≤ 1. By the assumption EXn 0 for n ≥ 1 and 3.18, we get
i
1
α
anj EXj I anj Xj < εn max
nα 1≤i≤n j1
≤
n n 1
anj EXj I anj Xj < εnα 1
anj EXj I anj Xj ≥ εnα nα j1
nα j1
≤
n 1 anj r EXj r ≤ Cnδ−αr −→ 0,
nαr j1
3.21
as n −→ ∞
following from δ < 1 and αr ≥ 1. We get the desired result from Theorem 3.2 immediately.
The proof is complete.
2
Theorem 3.4. Let {Xn , n ≥ 1} be a sequence of AANA random variables satisfying ∞
n1 q n < ∞,
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 any ε > 0, the following conditions are satisfied:
bn
∞
!
cn P |ani Xi | ≥ εbn1/t < ∞,
n1
∞
i1
bn
−p/t c n bn
|ani |p E|Xi |p I
n1
i1
!
3.22
|ani Xi | < εbn1/t < ∞,
then
⎧
⎫
∞
⎨
⎬
!%
i $
anj Xj − anj EXj I anj Xj < εbn1/t ≥ εbn1/t < ∞.
cn P max ⎩1≤i≤bn j1
⎭
n1
Proof. The proof is similar to that of Theorem 3.2, so we omit it.
3.23
Discrete Dynamics in Nature and Society
15
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 China Grant
nos. 10871001, 61075009, 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, Academic innovation team of
Anhui University KJTD001B, and Natural Science Research Project of Suzhou College
2009yzk25.
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