Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 863931, 24 pages
doi:10.1155/2012/863931
Research Article
On Complete Convergence of
Moving Average Process for AANA Sequence
Wenzhi Yang,1 Xuejun Wang,1 Nengxiang Ling,2 and Shuhe Hu1
1
2
School of Mathematical Science, Anhui University, Hefei 230039, China
School of Mathematics, Hefei University of Technology, Hefei 230009, China
Correspondence should be addressed to Shuhe Hu, hushuhe@263.net
Received 23 November 2011; Accepted 28 February 2012
Academic Editor: Cengiz Çinar
Copyright q 2012 Wenzhi Yang 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 investigate the moving average process such that Xn ∞
i1 ai Yin , n ≥ 1, where
i1 |ai | <
∞ and {Yi , 1 ≤ i < ∞} is a sequence of asymptotically almost negatively associated AANA
random variables. The complete convergence, complete moment convergence, and the existence of
the moment of supermum of normed partial sums are presented for this moving average process.
1. Introduction
We assume that {Yi , −∞ < i < ∞} is a doubly infinite sequence of identically distributed
random variables with E|Y1 | < ∞. Let {ai , −∞ < i < ∞} be an absolutely summable sequence
of real numbers and
Xn ∞
ai Yin ,
n≥1
1.1
i−∞
be the moving average process based on the sequence {Yi , −∞ < i < ∞}. As usual, Sn n
k1 Xk , n ≥ 1 denotes the sequence of partial sums.
Under the assumption that {Yi , −∞ < i < ∞} is a sequence of independent identically
distributed random variables, various results of the moving average process {Xn , n ≥ 1}
have been obtained. For example, Ibragimov 1 established the central limit theorem, Burton
and Dehling 2 obtained a large deviation principle, and Li et al. 3 gave the complete convergence result for {Xn , n ≥ 1}.
2
Discrete Dynamics in Nature and Society
Many authors extended the complete convergence of moving average process to the
case of dependent sequences, for example, Zhang 4 for ϕ-mixing sequence, Li and Zhang
5 for NA sequence. The following Theorems A and B are due to Zhang 4 and Kim et al.
6, respectively.
Theorem A. Suppose that {Yi , −∞ < i < ∞} is a sequence of identically distributed ϕ-mixing
1/2
m < ∞ and {Xn , n ≥ 1} is as in 1.1. Let hx > 0 x > 0 be a
random variables with ∞
m1 ϕ
slowly varying function and 1 ≤ p < 2, r ≥ 1. If Y1 0 and E|Y1 |rp h|Y1 |p < ∞, then
∞
nr−2 hnP |Sn | ≥ εn1/p < ∞,
∀ε > 0.
1.2
n1
Theorem B. Suppose that {Yi , −∞ < i < ∞} is a sequence of identically distributed ϕ-mixing
1/2
m < ∞ and {Xn , n ≥ 1} is as in 1.1.
random variables with EY1 0, EY12 < ∞ and ∞
m1 ϕ
Let hx > 0 x > 0 be a slowly varying function and 1 ≤ p < 2, r > 1. If E|Y1 |rp h|Y1 |p < ∞, then
∞
nr−2−1/p hnE |Sn | − εn1/p < ∞,
∀ε > 0,
1.3
n1
where x max{x, 0}.
Chen and Gan 7 investigated the moments of maximum of normed partial sums of
ρ-mixing random variables and gave the following result.
Theorem C. Let 0 < r < 2 and p > 0. Assume that {Xn , n ≥ 1} is a mean zero sequence of identically
2/s n
2 <
distributed ρ-mixing random variables with the maximal correlation coefficient rate ∞
n1 ρ
n
∞, where s 2 if p < 2 and s > p if p ≥ 2. Denote Sn i1 Xi , n ≥ 1. Then
⎧
⎪
E|X1 |r < ∞, if p < r,
⎪
⎪
⎨ E |X1 |r log1 |X1 | < ∞, if p r,
⎪
⎪
⎪
⎩
E|X1 |p < ∞, if p > r,
Xn p
< ∞,
E sup 1/r n≥1 n
S n p
E sup 1/r <∞
n≥1 n
1.4
are all equivalent.
Chen et al. 8 and Zhou 9 also studied limit behavior of moving average process
under ϕ-mixing assumption. For more related details of complete convergence, one can refer
to Hsu and Robbins 10, Chow 11, Shao 12, Li et al. 3, Zhang 4, Li and Zhang 5, Chen
and Gan 7, Kim et al. 6, Sung 13–15, Chen and Li 16, Zhou and Lin 17, and so forth.
Inspired by Zhang 4, Kim et al. 6, Chen and Gan 7, Sung 13–15, and other
papers above, we investigate the limit behavior of moving average process under AANA
Discrete Dynamics in Nature and Society
3
sequence, which is weaker than NA and obtain some similar results of Theorems A, B, and C.
The main results can be seen in Section 2 and their proofs are given in Section 3.
Recall that the sequence {Xn , n ≥ 1} is stochastically dominated by a nonnegative
random variable X if
sup P |Xn | > t ≤ CP X > t
for some positive constant C and ∀t ≥ 0.
n≥1
1.5
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,
1.6
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 NA.
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 1.7
for all n, k ≥ 1 and for all coordinate-wise nondecreasing continuous functions f and g whenever the variances exist.
The concept of NA sequence was introduced by Joag-Dev and Proschan 18. For the
basic properties and inequalities of NA sequence, one can refer to Joag-Dev and Proschan 18
and Matula 19. The family of AANA sequence contains NA in particular, independent
sequence 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 which is not
NA was constructed by Chandra and Ghosal 20, 21. For various results and applications
of AANA random variables can be found in Chandra Ghosal 21, Wang et al. 22, Ko et al.
23, Yuan and An 24, and Wang et al. 25, 26 among others.
For simplicity, in this paper we consider the moving average process:
Xn ∞
ai Yin ,
n ≥ 1,
1.8
i1
where
∞
i1 |ai | < ∞ and {Yi , 1 ≤ i < ∞} is a mean zero sequence of AANA random variables.
The following lemmas are our basic techniques to prove our results.
Lemma 1.3. Let {Xn , n ≥ 1} be a sequence of AANA random variables with mixing coefficients
{qn, n ≥ 1}. If f1 , f2 , . . . are all nondecreasing (or nonincreasing) continuous functions, then
{fn Xn , n ≥ 1} is still a sequence of AANA random variables with mixing coefficients {qn, n ≥
1}.
4
Discrete Dynamics in Nature and Society
Remark 1.4. Lemma 1.3 comes from Lemma 2.1 of Yuan and An 24, but the functions of
f1 , f2 , . . . in Lemma 2.1 of Yuan and An 24 are written to be all nondecreasing or nonincreasing functions. According to the definition of AANA, f1 , f2 , . . . should be all nondecreasing or nonincreasing continuous functions.
Lemma 1.5 cf. Wang et al. 25, Lemma 1.4. Let 1 < p ≤ 2 and {Xn , n ≥ 1} be a mean zero
2
sequence of AANA random variables with mixing coefficients {qn, n ≥ 1}. If ∞
n1 q n < ∞, then
there exists a positive constant Cp depending only on p such that
p k
n
≤ Cp E|Xi |p ,
E max Xi 1≤k≤n
i1
i1
for all n ≥ 1, where Cp 2p 22−p 6pp ∞
n1
1.9
q2 np−1 .
Lemma 1.6 cf. Wu 27, Lemma 4.1.6. Let {Xn , n ≥ 1} be a sequence of random variables, which
is stochastically dominated by a nonnegative random variable X. For any α > 0 and b > 0, the
following two statements hold:
E |Xn |α I|Xn | ≤ b ≤ C1 {EX α IX ≤ b bα P X > b},
E |Xn |α I|Xn | > b ≤ C2 EX α IX > b,
1.10
where C1 and C2 are positive constants.
Throughout the paper, IA is the indicator function of set A, x max{x, 0} and C,
C1 , C2 , . . . denote some positive constants not depending on n, which may be different in
various places.
2. The Main Results
Theorem 2.1. Let r > 1, 1 ≤ p < 2 and rp < 2. Assume that {Xn , n ≥ 1} is a moving average process
defined in 1.8, where {Yi , 1 ≤ i < ∞} is a mean zero sequence of AANA random variables with
∞ 2
rp
< ∞, then
n1 q n < ∞ and stochastically dominated by a nonnegative random variable Y . If EY
for every ε > 0,
∞
nr−2 P max |Sk | > εn1/p < ∞,
1≤k≤n
n1
∞
n1
Sk sup 1/p > ε < ∞.
k
2.1
r−2
n
P
k≥n
2.2
Discrete Dynamics in Nature and Society
5
Theorem 2.2. Let the conditions of Theorem 2.1 hold. Then for every ε > 0,
∞
nr−2−1/p E max |Sk | − εn1/p
< ∞,
1≤k≤n
n1
∞
Sk E sup 1/p − ε
< ∞.
k
2.3
r−2
n
2.4
k≥n
n1
Theorem 2.3. Let 0 < r < 2 and 0 < p < 2. Assume that {Xn , n ≥ 1} is a moving average process
defined in 1.8, where {Yi , 1 ≤ i < ∞} is a mean zero sequence of AANA random variables with
∞ 2
n1 q n < ∞ and stochastically dominated by a nonnegative random variable Y with EY < ∞.
Suppose that
⎧
⎪
⎪
⎪
⎪
⎪
for p < r,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
for p r,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪for p > r,
⎪
⎩
⎧ ⎨E Y log1 Y < ∞, if r 1,
⎩EY r < ∞,
⎧ E Y log1 Y < ∞,
⎪
⎪
⎪ ⎨
E Y log2 1 Y < ∞,
⎪
⎪
⎪
⎩ r
⎧E
Y log1 Y < ∞,
⎨E Y log1 Y < ∞,
⎩EY p < ∞,
if r > 1,
if 0 < r < 1,
if r 1,
2.5
if r > 1,
if p 1,
ifp > 1.
Then
S n p
< ∞.
E sup 1/r n
2.6
n≥1
3. The Proofs of Main Results
Proof of Theorem 2.1. Firstly, we show that the moving average process 1.8 converges almost
surely under the conditions of Theorem 2.1. Since rp > 1, it has EY < ∞, following from the
condition EY rp < ∞. On the other hand, by Lemma 1.6 with α 1 and b 1, one has
E|Yi | ≤ 1 C2 EY IY > 1 ≤ 1 C2 EY < ∞,
Consequently, by the condition
∞
i1
which implies
∞
i1
3.1
|ai | < ∞, we have that
∞
∞
E|ai Yin | ≤ C3 |ai | < ∞,
i1
1 ≤ i < ∞.
i1
ai Yin converges almost surely.
3.2
6
Discrete Dynamics in Nature and Society
Note that
n
Xk ∞
n ∞
in
ai Yik ai
Yk ,
k1 i1
k1
i1
n ≥ 1.
3.3
ki1
Since r > 1 and EY rp < ∞, one has EY p < ∞. Combining EY p < ∞ with EYj 0,
and Lemma 1.6, we can find that
∞
i1
|ai | < ∞
∞
ik n−1/p max ai
E Yj I Yj ≤ n1/p 1≤k≤n
i1 ji1
ik ∞ −1/p
1/p
n
max ai
E Yj I Yj > n
1≤k≤n
i1 ji1
≤ n−1/p
in ∞
E Yj I Yj > n1/p
|ai |
i1
≤ CE
3.4
1/p
n
ji1
p−1
1/p
YI Y > n
≤ CE Y p I Y > n1/p −→ 0 as n −→ ∞.
Meanwhile,
∞
ik n−1/p max ai
−n1/p E I Yj < −n1/p 1≤k≤n
i1 ji1
≤ n−1/p
in ∞
E Yj I Yj > n1/p ≤ CE Y p I Y > n1/p −→ 0 as n −→ ∞,
|ai |
i1
ji1
3.5
∞
ik
n−1/p max ai
n1/p E I Yj > n1/p 1≤k≤n
i1 ji1
≤ n−1/p
in ∞
E Yj I Yj > n1/p ≤ CE Y p I Y > n1/p −→ 0 as n −→ ∞.
|ai |
i1
ji1
Let
Ynj −n1/p I Yj < −n1/p Yj I Yj ≤ n1/p n1/p I Yj > n1/p ,
Ynj Ynj − EYnj ,
j ≥ 1.
j ≥ 1,
3.6
Discrete Dynamics in Nature and Society
7
Hence, for for all ε > 0, there exists an n0 such that
∞
ik
ε
n−1/p max ai
EYnj < ,
1≤k≤n
4
i1 ji1
n ≥ n0 .
3.7
Denote
∗
n1/p I Yj < −n1/p − n1/p I Yj > n1/p Yj I Yj > n1/p ,
Ynj
j ≥ 1.
3.8
∗
Noting that Yj Ynj
Ynj , j ≥ 1, we can find
⎞
⎛
∞
∞
∞
ik
1/p
εn
∗ ⎠
nr−2 P max |Sk | > εn1/p ≤
nr−2 P ⎝ max ai
Ynj
> 2
1≤k≤n
1≤k≤n
n1
n1
i1 ji1
⎛
⎞
∞
∞
ik
εn1/p
r−2 ⎝
⎠
Ynj >
n P max ai
2
1≤k≤n
n1
i1 ji1
⎛
⎞
∞
∞
ik
1/p
εn
∗ ⎠
nr−2 P ⎝ max ai
Ynj
≤
> 2
1≤k≤n
n1
i1 ji1
⎛
⎞
∞
ik
∞ εn1/p
r−2 ⎝
⎠
n P max ai
Ynj >
4
1≤k≤n
n1
i1 ji1
⎞
∞
ik
1/p
εn
⎠
nr−2 P ⎝ max ai
EYnj >
C
4
1≤k≤n
nn0
i1 ji1
∞
⎛
⎞
⎛
∞
∞
ik
1/p
εn
∗ ⎠
Ynj
≤ C nr−2 P ⎝ max ai
> 2
1≤k≤n
n1
i1 ji1
⎛
⎞
∞
∞
ik
1/p
εn
⎠
nr−2 P ⎝ max ai
Ynj >
4
1≤k≤n
n1
i1 ji1
: C I J.
3.9
8
Discrete Dynamics in Nature and Society
For I, by Markov’s inequality,
∞, it has
∞
∗
|ai | < ∞, |Ynj
| ≤ |Yj |I|Yj | > n1/p , Lemma 1.6 and EY rp <
i1
⎞
⎛
∞
∞
ik
2
∗ ⎠
nr−2 n−1/p E⎝ max ai
Ynj
I≤
ε n1
1≤k≤n
i1 ji1
≤ C1
∞
⎛
⎞
ik ∞
Yj I Yj > n1/p ⎠
nr−2 n−1/p |ai |E⎝ max
n1
≤ C2
∞
i1
1≤k≤n
ji1
nr−1−1/p E Y I Y > n1/p
n1
C2
∞
nr−1−1/p
∞
EY Im < Y p ≤ m 1
mn
n1
C2
∞
EY Im < Y p ≤ m 1
m1
≤ C3
∞
m
nr−1−1/p
n1
mr−1/p EY Im < Y p ≤ m 1 ≤ CEY rp < ∞.
m1
3.10
Since fj x −n1/p Ix < −n1/p xI|x| ≤ n1/p n1/p Ix > n1/p is a nondecreasing continuous function of x, we can find by using Lemma 1.3 that {Ynj , 1 ≤ j < ∞} is a mean zero AANA
2
2
2
sequence and EYnj
≤ EYnj
, Ynj
Yj2 I|Yj | ≤ n1/p n2/p I|Yj | > n1/p , j ≥ 1. Consequently, by
the property of AANA, Markov’s inequality, Hölder’s inequality, Lemma 1.5, Cr inequality,
and Lemma 1.6, we can check that
⎧
⎛
⎞2 ⎫
⎪
⎪
2 ∞
∞
ik
⎨
⎬
4
J≤
nr−2 n−2/p E max ⎝ ai
Ynj ⎠
⎪
⎪
ε n1
⎩1≤k≤n i1 ji1
⎭
⎧
⎞⎫2
⎛
⎬
2 ∞
∞ ik
⎨
4
≤
nr−2 n−2/p E
Ynj ⎠
|ai |1/2 ⎝|ai |1/2 max ⎩ i1
ε n1
1≤k≤n
⎭
ji1
⎧
⎛
⎞2 ⎫
2
⎪
⎪
2 ∞
∞
ik
⎬
⎨
4
r−2 −2/p
⎝
⎠
≤
n n
Ynj
|ai | sup E max
⎪
⎪
ε n1
⎭
⎩1≤k≤n ji1
i≥1
i1
≤ C1
∞
n1
≤ C2
∞
n1
nr−2 n−2/p sup
in
2
EYnj
i≥1 ji1
nr−2 n−2/p sup
in ! "
E Yj2 I Yj ≤ n1/p n2/p E I Yj > n1/p
i≥1 ji1
Discrete Dynamics in Nature and Society
≤ C3
∞
9
∞
nr−1−2/p E Y 2 I Y ≤ n1/p C4 nr−1 P Y > n1/p
n1
≤ C3
∞
n1
∞
nr−1−2/p E Y 2 I Y ≤ n1/p C4 nr−1−1/p E Y I Y > n1/p
n1
n1
: C3 J1 C4 J2 .
3.11
Since rp < 2, it can be seen by EY rp < ∞ that
J1 ∞
n
nr−1−2/p E Y 2 I i − 11/p < Y ≤ i1/p
n1
i1
∞ ∞
E Y 2 I i − 11/p < Y ≤ i1/p
nr−1−2/p
≤ C1
3.12
ni
i1
∞ E Y rp Y 2−rp I i − 11/p < Y ≤ i1/p ir−2/p ≤ C2 EY rp < ∞.
i1
By the proof of 3.10, we have J2 ≤ CEY rp < ∞. Therefore, 2.1 follows from 3.9, 3.10,
3.11, and 3.12.
Inspired by the proof of Theorem 12.1 of Gut 28, it can be checked that
∞
r−2
n
Sk 2/p
sup 1/p > 2 ε
k
P
k≥n
n1
m
∞ 2
−1
r−2
n
m1 n2m−1
≤ 22−r
∞
P
≤2
∞
P
k≥n
m
2
−1
Sk sup 1/p > 22/p ε
2mr−2
m−1 k
k≥2
m1
2−r
mr−1
2
P
2
∞
∞
mr−1
2
P
22−r
Sk 2/p
sup max 1/p > 2 ε
2l−1 ≤k<2l k
l≥m
2mr−1
m1
Sk 2/p
sup 1/p > 2 ε
m−1 k
m1
≤ 22−r
n2m−1
k≥2
m1
2−r
Sk 2/p
sup 1/p > 2 ε
k
∞ P max |Sk | > ε2l1/p
lm
1≤k≤2l
l
∞ P max |Sk | > ε2l1/p
2mr−1
l1
1≤k≤2l
m1
10
Discrete Dynamics in Nature and Society
≤ C1
∞
2lr−1 P max |Sk | > ε2l1/p
1≤k≤2l
l1
22−r C1
l1
−1
∞ 2
2l1r−2 P max |Sk | > ε2l1/p
1≤k≤2l
l1 n2l
≤ 22−r C1
l1
−1
∞ 2
nr−2 P max |Sk | > εn1/p
since r < 2
1≤k≤n
l1 n2l
≤ 22−r C1
∞
nr−2 P max |Sk | > εn1/p .
1≤k≤n
n1
3.13
Combining 2.1 with the inequality above, we obtain 2.2 immediately.
Proof of Theorem 2.2. For all ε > 0, it has
∞
nr−2−1/p E max |Sk | − εn1/p
1≤k≤n
n1
#∞ ∞
nr−2−1/p
P max |Sk | − εn1/p > t dt
n1
1≤k≤n
0
# n1/p ∞
r−2−1/p
1/p
n
P max |Sk | − εn > t dt
n1
3.14
1≤k≤n
0
#∞ ∞
r−2−1/p
1/p
P max |Sk | − εn > t dt
n
n1
n1/p
1≤k≤n
#∞ ∞
∞
r−2
1/p
r−2−1/p
n P max |Sk | > εn
P max |Sk | > t dt.
n
≤
n1
1≤k≤n
n1
n1/p
1≤k≤n
By Theorem 2.1, in order to prove 2.3, we only have to prove that
∞
n1
r−2−1/p
#∞
n
n1/p
P max |Sk | > t dt < ∞.
3.15
1≤k≤n
For t > 0, let
Ytj −tI Yj < −t Yj I Yj ≤ t tI Yj > t ,
Ytj Ytj − EYtj , j ≥ 1,
Ytj∗ tI Yj < −t − tI Yj > t Yj I Yj > t ,
j ≥ 1,
3.16
j ≥ 1.
Discrete Dynamics in Nature and Society
11
Since Yj Ytj∗ Ytj , j ≥ 1, it is easy to see that
∞
nr−2−1/p
#∞
n1/p
n1
P max |Sk | > t dt
1≤k≤n
⎞
⎛
#∞
∞
∞
ik
t
nr−2−1/p
P ⎝ max ai
Ytj∗ > ⎠dt
≤
1≤k≤n
1/p
2
n
n1
i1 ji1
⎞
⎛
#∞
∞
∞
ik
t
r−2−1/p
n
P ⎝ max ai
Ytj > ⎠dt
1≤k≤n
4
n1/p
n1
i1 ji1
3.17
⎞
⎛
#∞
∞
∞
ik
t
nr−2−1/p
P ⎝ max ai
EYtj > ⎠dt
1≤k≤n
4
n1/p
n1
i1 ji1
: I1 I2 I3 .
For I1 , by Markov’s inequality, |Ytj∗ | ≤ |Yj |I|Yj | > t, Lemma 1.6 and EY rp < ∞, we get that
⎞
⎛
#∞
∞
∞
ik
r−2−1/p
−1 ⎝
∗ ⎠
I1 ≤ 2 n
t E max ai
Ytj dt
1≤k≤n
n1/p
n1
i1 ji1
⎞
⎛
#∞
∞
ik ∞
Yj I Yj > t ⎠dt
≤ 2 nr−2−1/p
t−1 |ai |E⎝ max
n1/p
n1
≤ C1
∞
nr−1−1/p
∞
∞
nr−1−1/p
∞
∞
∞
t−1 E Y I Y > m1/p dt
m1/p
m1/p−1−1/p E Y I Y > m1/p
mn
m
m−1 E Y I Y > m1/p
nr−1−1/p
m1
≤ C3
∞ # m1
mn
n1
C2
t−1 EY IY > tdt
1/p
nr−1−1/p
n1
≤ C2
#∞
n1/p
n1
C1
1≤k≤n
ji1
i1
n1
mr−1−1/p E Y I Y > m1/p ≤ C4 EY rp < ∞.
m1
3.18
12
Discrete Dynamics in Nature and Society
From the fact that {Ytj , 1 ≤ j < ∞} is a mean zero AANA sequence and EYtj2 ≤ EYtj , Ytj2 Yj2 I|Yj | ≤ t t2 I|Yj | > t, j ≥ 1, similar to the proof of 3.11, we have
I2 ≤ 42
∞
nr−2−1/p
n1/p
n1
≤ 42
∞
nr−2−1/p
#∞
≤ C1
r−2−1/p
≤ C2
∞
∞
n1/p
r−2−1/p
#∞
n1/p
nr−1−1/p
#∞
n1/p
n1
|ai |
in
i1
ji1
⎧
⎪
⎨
⎞2 ⎫
⎪
⎬
sup E max ⎝
Ytj ⎠ dt
⎪
⎪
⎩1≤k≤n ji1
⎭
i≥1
⎛
ik
EYtj2 dt
3.19
i≥1 ji1
t−2 sup
n
2
i1
t−2 sup
n
n1
≤ C3
t−2
#∞
n1
∞
1≤k≤n
n1/p
n1
∞
⎛
⎞2 ⎞
∞
ik
⎟
⎜
t−2 E⎝max ⎝ ai
Ytj ⎠ ⎠dt
⎛
#∞
in ! "
dt
E Yj2 I Yj ≤ t t2 E I Yj > t
i≥1 ji1
#∞
∞
t−2 E Y 2 IY ≤ t dt C4 nr−1−1/p
P Y > tdt
n1
n1/p
: C3 I21 C4 I22 .
It follows from rp < 2 and EY rp < ∞ that
I21 ∞
n
mn
n1
≤ C1
∞
nr−1−1/p
∞
m1/p
∞
m1/p−1−2/p E Y 2 I Y ≤ m 11/p
m
m−1/p−1 E Y 2 I Y ≤ m 11/p
nr−1−1/p
m1
≤ C2
∞
t−2 E Y 2 IY ≤ t dt
mn
n1
C1
∞ # m1
1/p
r−1−1/p
n1
mr−1−2/p E Y 2 I Y ≤ m 11/p
m1
C2
∞
mr−1−2/p
m1
C2
∞
E Y 2 I i − 11/p < Y ≤ i1/p
m1
i1
mr−1−2/p E Y 2 I m1/p < Y ≤ m 11/p
m1
C2
∞
mr−1−2/p
m1
C2
∞
m E Y 2 I i − 11/p < Y ≤ i1/p
i1
mr−1−2/p E Y rp Y 2−rp I m1/p < Y ≤ m 11/p
m1
Discrete Dynamics in Nature and Society
C2
13
∞
∞ E Y 2 I i − 11/p < Y ≤ i1/p
mr−1−2/p
mi
i1
∞
≤ 22−rp/p C2
m−1 E Y rp I m1/p < Y ≤ m 11/p
m1
∞ C3 E Y rp Y 2−rp I i − 11/p < Y ≤ i1/p ir−2/p ≤ C4 EY rp < ∞.
i1
3.20
By the proof of 3.18, one has
I22 ≤
∞
nr−1−1/p
n1
#∞
n1/p
t−1 EY IY > tdt ≤ CEY rp < ∞.
3.21
On the other hand, by the property EYj 0, we have
ik
∞ max ai
EYtj 1≤k≤n
i1 ji1
ik
∞ max ai
E Yj I Yj ≤ t − tE I Yj < −t tE I Yj > t 1≤k≤n
i1 ji1
ik
∞ max ai
E Yj I Yj > t tE I Yj < −t − tE I Yj > t 1≤k≤n
i1 ji1
∞
in
≤ 2 |ai |
E Yj I Yj > t .
i1
ji1
3.22
Thus, by Lemma 1.6 and the proof of 3.18, it can be seen that
⎞
⎛
#∞
∞
∞
ik
I3 ≤ 4 nr−2−1/p
t−1 ⎝ max ai
EYtj ⎠dt
1≤k≤n
n1/p
n1
i1 ji1
#∞
∞
in
∞
≤ 8 nr−2−1/p
t−1 |ai |
E Yj I Yj > t dt
n1
n1/p
i1
3.23
ji1
#∞
∞
r−1−1/p
≤ C1 n
t−1 EY IY > tdt ≤ C2 EY rp < ∞.
n1
n1/p
Consequently, by 3.14, 3.17, 3.18, 3.19, 3.20, 3.21, and 3.23 and Theorem 2.1, 2.3
holds true.
14
Discrete Dynamics in Nature and Society
Next, we prove 2.4. It is easy to see that
∞
Sk r−2
2/p
n E sup 1/p − ε2
k≥n k
n1
∞
#∞ Sk 2/p
P sup 1/p > ε2 t dt
0
k≥n k
r−2
n
n1
m
∞ 2
−1
r−2
n
m1n2m−1
≤2
2−r
∞ #∞
≤2
∞
P
0
m1
2−r
#∞ Sk 2/p
P sup 1/p > ε2 t dt
0
k≥n k
k≥2
2
∞
#∞ Sk 2/p
P sup 1/p > ε2 t dt
0
k≥2m−1 k
mr−1
#∞ Sk 2/p
P sup max 1/p > ε2 t dt
l−1
l
0
l≥m 2 ≤k<2 k
2
2
m1
≤ 22−r
∞
2mr−1
m1
22−r
n2m−1
mr−1
m1
2−r
m
2
−1
Sk 2/p
sup 1/p > ε2 t dt
2mr−2
m−1 k
∞ #∞ P max |Sk | > ε22/p t 2l−1/p dt
0
lm
1≤k≤2l
l
∞ #∞ P max |Sk | > ε22/p t 2l−1/p dt 2mr−1
l1
0
1≤k≤2l
m1
#∞ ∞
2−r
lr−1
≤2
2
P max |Sk | > ε22/p t 2l−1/p dt
≤ C1
let s 2l−1/p t
#∞ ∞
2lr−1−1/p
P max |Sk | > ε2l1/p s ds
0
l1
21/p−r
2
1≤k≤2l
0
l1
1≤k≤2l
#∞ l1
−1
∞ 2
l1r−2−1/p
l1/p
C1
2
P max |Sk | > ε2
s ds
≤ 221/p−r C1
#∞ l1
−1
∞ 2
nr−2−1/p
P max |Sk | > εn1/p s ds since r < 2
l1 n2l
≤ 221/p−r C1
1≤k≤2l
0
l1 n2l
0
1≤k≤n
∞
nr−2−1/p E max |Sk | − εn1/p
< ∞.
n1
1≤k≤n
3.24
Therefore, 2.4 holds true following from 2.3.
Discrete Dynamics in Nature and Society
15
Proof of Theorem 2.3. Similar to the proof of Theorem 2.1, by
∞
i1 ai Yin converges almost surely. It can be seen that
∞
i1
|ai | < ∞ and EY < ∞,
#∞ S n p
Sn 1/p
dt
P sup 1/r > t
E sup 1/r n
n
0
n≥1
≤2
p/r
≤2
p/r
≤ 2p/r
≤ 2p/r
n≥1
Sn 1/p
dt
sup 1/r > t
n
#∞
P
2p/r
n≥1
Sn 1/p
dt
sup max 1/r > t
2k−1 ≤n<2k n
#∞
P
2p/r
k≥1
#∞ ∞ Sn P
max 1/r > t1/p dt
2k−1 ≤n<2k n
2p/r k1
#∞ ∞ P max |Sn | > 2k−1/r t1/p dt
let s 2k−1p/r t
2p/r k1
2p/r 2p/r
∞
3.25
1≤n≤2k
2−kp/r
#∞
2kp/r
k1
P
max |Sn | > s1/p ds.
1≤n≤2k
For s1/p > 0, let
Ysj −s1/p I Yj < −s1/p Yj I Yj ≤ s1/p s1/p I Yj > s1/p ,
Ysj Ysj − EYsj , j ≥ 1,
Ysj∗ s1/p I Yj < −s1/p − s1/p I Yj > s1/p Yj I Yj > s1/p ,
j ≥ 1,
3.26
j ≥ 1.
Since Yj Ysj∗ Ysj , j ≥ 1, then
∞
k1
2−kp/r
#∞
P
2kp/r
max |Sn | > s1/p ds
1≤n≤2k
⎛
⎞
#∞
∞
∞
in
s1/p
−kp/r
∗
⎠ds
≤
2
P ⎝ max ai
Ysj >
2
1≤n≤2k i1 ji1
2kp/r
k1
⎞
⎛
#∞
∞
∞
in
1/p
s
⎠ds
2−kp/r
P ⎝ max ai
Ysj >
k
4
kp/r
1≤n≤2
2
i1 ji1
k1
⎛
⎞
#∞
∞
∞
in
1/p
s
⎠ds
2−kp/r
P ⎝ max ai
EYsj >
k
4
kp/r
1≤n≤2
2
i1 ji1
k1
: H1 H2 H3 .
3.27
16
Discrete Dynamics in Nature and Society
For H1 , similar to 3.18, by Markov’s inequality and Lemma 1.6, one has
⎞
⎛
#∞
∞
∞
in
H1 ≤ 2 2−kp/r
s−1/p E⎝ max ai
Ysj∗ ⎠ds
k
1≤n≤2 i1 ji1
2kp/r
k1
⎞
⎛
#∞
∞
in ∞
−kp/r
−1/p
1/p
Yj I Yj > s
⎠ds
≤2 2
s
|ai |E⎝ max
2kp/r
k1
i1
1≤n≤2k ji1
#∞
∞
k−kp/r
≤ C1 2
s−1/p E Y I Y > s1/p ds
2kp/r
k1
∞ # 2m1p/r
∞
k−kp/r
C1 2
s−1/p E Y I Y > s1/p ds
k1
mk
2mp/r
3.28
∞
∞
≤ C2 2k−kp/r
2mp/r−m/r E Y I Y > 2m/r
k1
C2
∞
mk
m
2mp/r−m/r E Y I Y > 2m/r
2k−kp/r
m1
k1
⎧ ∞
⎪
⎪
2m−m/r E Y I Y > 2m/r ,
if p < r,
C
⎪
3
⎪
⎪
⎪
m1
⎪
⎪ ∞
⎪
⎨ ≤ C4 m2m−m/r E Y I Y > 2m/r , if p r,
⎪
⎪
m1
⎪
⎪
⎪
∞
⎪
⎪
⎪
⎪
⎩C5 2mp/r−m/r E Y I Y > 2m/r , if p > r.
m1
For the case p < r, if 0 < r < 1, then
∞
∞
∞
2m−m/r E Y I Y > 2m/r 2m−m/r
E Y I 2k/r < Y ≤ 2k1/r
m1
m1
km
∞ k
E Y I 2k/r < Y ≤ 2k1/r
2m1−1/r
k1
≤ C1
m1
∞ E Y I 2k/r < Y ≤ 2k1/r
k1
≤ C1 EY.
3.29
Discrete Dynamics in Nature and Society
17
If r 1, then
∞
∞
2m−m/r E Y I Y > 2m/r EY IY > 2m m1
m1
∞
∞ E Y I 2k < Y ≤ 2k1
m1 km
∞ k
E Y I 2k < Y ≤ 2k1
1
m1
k1
3.30
∞
kE Y I 2k < Y ≤ 2k1
k1
≤ C1
∞ E Y log1 Y I 2k < Y ≤ 2k1
k1
≤ C1 E Y log1 Y .
Otherwise for r > 1, it has
∞
∞
∞
2m−m/r E Y I Y > 2m/r 2m−m/r
E Y I 2k/r < Y ≤ 2k1/r
m1
m1
km
∞ k
E Y I 2k/r < Y ≤ 2k1/r
2m−m/r
m1
k1
≤ C1
∞
2k−k/r E Y I 2k/r < Y ≤ 2k1/r
3.31
k1
≤ C1
∞ E Y r I 2k/r < Y ≤ 2k1/r ≤ C1 EY r .
k1
Similarly, for the case p r, if 0 < r < 1, then
∞
∞
∞
m2m−m/r E Y I Y > 2m/r m2m−m/r
E Y I 2k/r < Y ≤ 2k1/r
m1
m1
km
∞ k
E Y I 2k/r < Y ≤ 2k1/r
m2m1−1/r
m1
k1
≤
∞ k
E Y I 2k/r < Y ≤ 2k1/r k 2m1−1/r
m1
k1
≤ C1
∞
k1
kE Y I 2k/r < Y ≤ 2k1/r
≤ C2 E Y log1 Y .
3.32
18
Discrete Dynamics in Nature and Society
If r 1, then
∞
∞
∞
m2m−m/r E Y I Y > 2m/r m E Y I 2k < Y ≤ 2k1
m1
m1
km
∞ k
E Y I 2k < Y ≤ 2k1
m
m1
k1
≤ C2
∞
k2 E Y I 2k < Y ≤ 2k1
3.33
k1
≤ C2
∞ E Y log2 1 Y I 2k < Y ≤ 2k1
k1
≤ C2 E Y log2 1 Y .
Otherwise, for r > 1, it follows
∞
m2m−m/r E Y I Y > 2m/r
m1
∞
m2m−m/r
m1
∞
E Y I 2k/r < Y ≤ 2k1/r
km
∞ k
E Y I 2k/r < Y ≤ 2k1/r
m2m−m/r
≤
3.34
m1
k1
∞ k
E Y I 2k/r < Y ≤ 2k1/r k 2m−m/r
m1
k1
≤ C1
∞
k2k−k/r E Y I 2k/r < Y ≤ 2k1/r ≤ C2 E Y r log1 Y .
k1
On the other hand, for the case p > r, if 0 < p < 1, then
∞
∞
∞
2mp/r−m/r E Y I Y > 2m/r 2mp−1/r
E Y I 2k/r < Y ≤ 2k1/r
m1
m1
km
∞ k
E Y I 2k/r < Y ≤ 2k1/r
2mp−1/r
k1
≤ C1
m1
∞ E Y I 2k/r < Y ≤ 2k1/r
k1
≤ C1 EY.
3.35
Discrete Dynamics in Nature and Society
19
If p 1, then
∞
∞ 2mp/r−m/r E Y I Y > 2m/r E Y I 2k/r < Y ≤ 2k1/r
∞
m1
m1km
∞ k
E Y I 2k/r < Y ≤ 2k1/r
1
m1
k1
∞
kE Y I 2k/r < Y ≤ 2k1/r
k1
≤ C1 EY log1 Y .
3.36
For p > 1, it has
∞
∞
∞
2mp/r−m/r E Y I Y > 2m/r 2mp−1/r
E Y I 2k/r < Y ≤ 2k1/r
m1
m1
km
∞ k
E Y I 2k/r < Y ≤ 2k1/r
2mp−1/r
m1
k1
≤ C1
∞
2kp−1/r E Y I 2k/r < Y ≤ 2k1/r
k1
≤ C1
∞ E Y p I 2k/r < Y ≤ 2k1/r ≤ C1 EY p .
k1
3.37
Consequently, by 3.28, the conditions of Theorem 2.3 and inequalities above, we obtain that
⎧ ∞
⎪
m−m/r
⎪
E Y I Y > 2m/r ,
C
⎪ 1 2
⎪
⎪
⎪
m1
⎪
⎪
⎪
∞
⎨ H1 ≤ C2 m2m−m/r E Y I Y > 2m/r ,
⎪
⎪
m1
⎪
⎪
⎪
∞
⎪
⎪
⎪C 2mp/r−m/r E
Y I Y > 2m/r ,
⎪
⎩ 3
m1
if p < r,
if p r,
if p > r
20
Discrete Dynamics in Nature and Society
⎧
⎧
⎪
⎪
if 0 < r < 1,
C4 EY < ∞,
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
for p < r, C5 E Y log1 Y < ∞, if r 1,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
if r > 1,
C EY r < ∞,
⎪
⎪
⎧ 6 ⎪
⎪
⎪
C7 E Y log1 Y < ∞, if 0 < r < 1,
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎨
≤ for p r, C8 E Y log2 1 Y < ∞, if r 1,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
C9 E Y r log1 Y < ∞, if r > 1,
⎪
⎪
⎧
⎪
⎪
⎪
⎪
if 0 < p < 1,
C10 EY < ∞,
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
for p > r, C11 E Y log1 Y < ∞, if p 1,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎩
if p > 1.
C12 EY p < ∞,
3.38
Since {Ysj , 1 ≤ j < ∞} is a mean zero AANA sequence and EYsj2 ≤ EYsj2 , Ysj2 Yj2 I|Yj | ≤
s1/p s2/p I|Yj | > s1/p , j ≥ 1, similar to the proof of 3.11, we obtain that
⎧
⎛
⎞2 ⎫
⎪
⎪
#∞
∞
∞
in
⎨
⎬
H2 ≤ C1 2−kp/r
s−2/p E max ⎝ ai
Ysj ⎠ ds
⎪
⎪
⎩1≤n≤2k i1 ji1
⎭
2kp/r
k1
⎧
⎛
⎞2 ⎫
2
⎪
⎪
#∞
∞
in
∞
⎨
⎬
−kp/r
−2/p
⎝
⎠
≤ C1 2
ds
s
Ysj
|ai | supE max
⎪
⎩1≤n≤2k ji1
⎭
2kp/r
i≥1 ⎪
i1
k1
#∞
i2
∞
k
−kp/r
−2/p
≤ C2 2
s
sup
EYsj2 ds
2kp/r
k1
≤ C3
∞
2−kp/rk
C4
#∞
2kp/r
k1
3.39
i≥1 ji1
s−2/p E Y 2 I Y ≤ s1/p ds
#∞
∞
2−kp/rk
P Y > s1/p ds
2kp/r
k1
: C3 H21 C4 H22 .
Similar to the proof of Theorem 1.1 of Chen and Gan 7, by p < 2 and the conditions of
Theorem 2.3, we have that
H21 ∞
−kp/rk
2
k1
≤
∞
k1
∞ # 2m1p/r
mk
2−kp/rk
∞
mk
2mp/r
s−2/p E Y 2 I Y ≤ s1/p ds
2mp/r−2m/r E Y 2 I Y ≤ 2m1/r
Discrete Dynamics in Nature and Society
∞
21
m
2mp−2/r E Y 2 I Y ≤ 2m1/r
2k1−p/r
m1
k1
⎧ ∞
⎪
⎪
⎪
C1 2mr−2/r E Y 2 I Y ≤ 2m1/r ,
⎪
⎪
⎪
⎪
⎪ m1
⎪
∞
⎨ ≤ C2 m2mr−2/r E Y 2 I Y ≤ 2m1/r ,
⎪
⎪
m1
⎪
⎪
⎪
∞
⎪
⎪
mp−2/r
⎪
C
E Y 2 I Y ≤ 2m1/r ,
⎪
⎩ 3 2
if p < r,
if p r,
if p > r
m1
⎧
⎪
C4 EY r < ∞,
⎪
⎪
⎨
≤ C5 E Y r log1 Y < ∞,
⎪
⎪
⎪
⎩
C6 EY p < ∞,
if p < r,
if p r,
if p > r.
3.40
On the other hand, by the proof of 3.28 and 3.38, it follows
H22 ≤
∞
2k−kp/r
#∞
s−1/p E Y I Y > s1/p ds
2kp/r
k1
⎧
⎧
⎪
⎪
C1 EY < ∞,
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
for p < r, C2 E Y log1 Y < ∞,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
C3 EY r < ∞,
⎪
⎪
⎧
⎪
⎪
⎪
C4 E Y log1 Y < ∞,
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎨
≤ for p r, C5 E Y log2 1 Y < ∞,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
C6 E Y r log1 Y < ∞,
⎪
⎪
⎧
⎪
⎪
⎪
⎪
C7 EY < ∞,
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
for p > r, C8 E Y log1 Y < ∞,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎩
C9 EY p < ∞,
if 0 < r < 1,
if r 1,
if r > 1,
if 0 < r < 1,
3.41
if r 1,
if r > 1,
if 0 < p < 1,
if p 1,
if p > 1.
Similar to the proof of 3.23, by the property EYj 0 and the proofs of 3.28 and 3.38, one
has
⎞
⎛
#∞
∞
in
∞ −kp/r
−1/p ⎝
max ai
H3 ≤ 4 2
s
EYsj ⎠ds
1≤n≤2k i1 ji1
2kp/r
k1
⎛
⎞
#∞
∞
∞
in ≤ 8 2−kp/r
s−1/p ⎝ max |ai |
E Yj I Yj > s1/p ⎠ds
k1
2kp/r
1≤n≤2k i1
ji1
22
Discrete Dynamics in Nature and Society
≤ C1
#∞
∞
2k−kp/r
s−1/p E Y I Y > s1/p ds
k1
2kp/r
⎧
⎧
⎪
⎪
C1 EY < ∞,
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
for p < r, C2 E Y log1 Y < ∞,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
C EY r < ∞,
⎪
⎪
⎧ 3 ⎪
⎪
⎪
C4 E Y log1 Y < ∞,
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎨
≤ for p r, C5 E Y log2 1 Y < ∞,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
C6 E Y r log1 Y < ∞,
⎪
⎪
⎧
⎪
⎪
⎪
⎪
C7 EY < ∞,
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
for p > r, C8 E Y log1 Y < ∞,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎩
C9 EY p < ∞,
if 0 < r < 1,
if r 1,
if r > 1,
if 0 < r < 1,
if r 1,
if r > 1,
if 0 < p < 1,
if p 1,
if p > 1.
3.42
Consequently, by 3.25, 3.27, 3.28, 3.38, 3.39, 3.40, 3.41, and 3.42, we finally
obtain 2.6.
Remark 3.1. Zhou and Lin 17 obtained the result 2.6 for partial sums of moving average
process under ϕ-mixing sequence. But there is one problem in their proof. On page 694 of
Zhou and Lin 17, they presented that
⎧ ∞
⎪
⎪
⎪
if p < r,
C 2m−m/r E |Y1 |I |Y1 | > 2m/r ,
⎪
⎪
⎪
m1
⎪
⎪
⎪
∞
⎨ I1 ≤ · · · ≤ C m2m−m/r E |Y1 |I |Y1 | > 2m/r , if p r,
⎪
⎪
m1
⎪
⎪
⎪
∞
⎪
⎪
⎪
⎪C 2mp/r−m/r E |Y1 |I |Y1 | > 2m/r , if p > r,
⎩
m1
⎧
⎪
CE|Y1 |r < ∞,
⎪
⎪
⎨
≤ CE |Y1 |r log1 |Y1 | < ∞,
⎪
⎪
⎪
⎩
CE|Y1 |p < ∞,
3.43
if p < r,
if p r,
if p > r,
where 1 ≤ r < 2 and p > 0. For the case p < r, by taking r 1, we cannot get that
∞
∞
2m−m/r E |Y1 |I |Y1 | > 2m/r E|Y1 |I|Y1 | > 2m ≤ CE|Y1 | < ∞.
m1
m1
3.44
Discrete Dynamics in Nature and Society
23
Here, we give a counter example to illustrate this problem. Assume that the density function
of nonnegative random variable Y is
f y C
y2 ln2 y
y > 2, C ,
&# ∞
2
'−1
1
y2 ln2 y
dy
.
3.45
Obviously, it can be found that
EY C
#∞
2
1
y ln2 y
dy < ∞.
3.46
But for r 1,
∞
∞
∞ ∞ n
2m−m/r E Y I Y > 2m/r E Y I 2n < Y ≤ 2n1 E Y I 2n < Y ≤ 2n1
1
m1
m1 nm
C
n1
# 2n1
∞
n
n1
2n
1
yln2 y
dy ∞
1
C ∞.
ln 2 n1 n 1
m1
3.47
Acknowledgments
The authors are deeply grateful to Editor Cengiz Çinar and two anonymous referees for
careful reading of the paper and valuable suggestions which helped in improving an earlier
version of this paper. This work is supported by the NNSF of China 11171001, HSSPF
of the Ministry of Education of China 10YJA910005, Mathematical Tianyuan Youth Fund
11126176, Provincial Natural Science Research Project of Anhui Colleges KJ2010A005,
Natural Science Foundation of Anhui Province 1208085QA03, and Academic Innovation
Team of Anhui University KJTD001B.
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