Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 128492, 16 pages
doi:10.1155/2012/128492
Research Article
Complete Convergence for Moving Average Process
of Martingale Differences
Wenzhi Yang, Shuhe Hu, and Xuejun Wang
School of Mathematical Science, Anhui University, Hefei 230039, China
Correspondence should be addressed to Shuhe Hu, hushuhe@263.net
Received 9 March 2012; Accepted 14 May 2012
Academic Editor: Chuanxi Qian
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.
Under some simple conditions, by using some techniques such as truncated method for random
variables see e.g., Gut 2005 and properties of martingale differences, we studied the moving
process based on martingale differences and obtained complete convergence and complete
moment convergence for this moving process. Our results extend some related ones.
1. Introduction
Let {Yi , −∞ < i < ∞} be a doubly infinite sequence of random variables. Assume that
{ai , −∞ < i < ∞} is an absolutely summable sequence of real numbers and
Xn ∞
ai Yin ,
n≥1
1.1
i−∞
is the moving average process based on the sequence {Yi , −∞ < i < ∞}. As usual, Sn nk1 Xk ,
n ≥ 1, denotes the sequence of partial sums.
For the moving average process {Xn , n ≥ 1}, where {Yi , −∞ < i < ∞} is a sequence of
independent identically distributed i.i.d. random variables, 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}. Zhang 4 and Li and Zhang
5 extended the complete convergence of moving average process for i.i.d. sequence to ϕmixing sequence and NA sequence, respectively. Theorems A and B are due to Zhang 4
and Kim et al. 6, respectively.
2
Discrete Dynamics in Nature and Society
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 et al. 7 and Zhou 8 also studied the limit behavior of moving average process
under ϕ-mixing assumption. Yang et al. 9 investigated the moving average process for
AANA sequence. For more works on complete convergence, one can refer to 3–6, 10–13
and the references therein.
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, ∀t ≥ 0.
n≥1
1.4
Recently, Chen and Li 14 investigated the limit behavior of moving process under
martingale difference sequences. They obtained the following theorems.
Theorem C. Let r ≥ 1, 1 ≤ p < 2 and rp < 2. Assume that {Xn , n ≥ 1} is a moving average process
defined in 1.1, where {Yi , Fi , −∞ < i < ∞} is a martingale difference related to an increasing
sequence of σ-fields Fi and stochastically dominated by a nonnegative random variable Y . If EY rp Y log1 Y < ∞, then for every ε > 0,
∞
n1
r−2
n
P
max |Sk | > εn
1/p
1≤k≤n
< ∞.
1.5
Theorem D. Let r ≥ 1, 1 ≤ p < 2, rp < 2 and 0 < q < 2. Assume that {Xn , n ≥ 1} is a moving
average process defined in 1.1, where {Yi , Fi , −∞ < i < ∞} is a martingale difference related to an
increasing sequence of σ-fields Fi and stochastically dominated by a nonnegative random variable Y .
If
⎧ ⎪
E Y rp Y log1 Y < ∞,
⎪
⎪
⎨ E Y rp log 1 Y Y log2 1 Y < ∞,
⎪
⎪
⎪
⎩ q
EY < ∞,
if q < rp,
if q rp,
if q > rp,
1.6
Discrete Dynamics in Nature and Society
3
then for every ε > 0,
∞
q
nr−2−q/p E max |Sk | − εn1/p
< ∞,
n1
1≤k≤n
1.7
q
where x x when x > 0 and x 0 when x ≤ 0 and x x q .
Inspired by Chen and Li 14, Chen et al. 7, Sung 13 and other papers above,
we go on to investigate the limit behavior of moving process under martingale difference
sequence and obtain some similar results of Theorems C and D, but we only need some
simple conditions. Our results extend some results of Chen and Li 14 see Remark 3.3 in
Section 3. Two lemmas and two theorems are given in Sections 2 and 3, respectively. The
proofs of theorems are presented in Section 4.
For various results of martingales, one can refer to Chow 15, Hall and Heyde 16,
Yu 17, Ghosal and Chandra 18, and so forth. As an application of moving average
process based on martingale differences, we can refer to 19–22 and the references therein.
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. Two Lemmas
The following lemmas are our basic techniques to prove our results.
Lemma 2.1 cf. Hall and Heyde 16, Theorem 2.11. If {Xi , Fi , 1 ≤ i ≤ n} is a martingale
difference and p > 0, then there exists a constant C depending only on p such that
⎧ ⎫
p ⎬
k
n
⎨
p/2
.
≤C E
E Xi2 | Fi−1
E max |Xi |p
E max Xi ⎩
⎭
1≤i≤n
1≤k≤n i1
i1
2.1
Lemma 2.2 cf. Wu 23, Lemma 4.1.6. Let {Xn , n ≥ 1} be a sequence of random variables, which
is stochastically dominated by a nonnegative random variable X. Then for any a > 0 and b > 0, the
following two statements hold:
E |Xn |a I|Xn | ≤ b ≤ C1 {EX a IX ≤ b ba P X > b},
E |Xn |a I|Xn | > b ≤ C2 EX a IX > b,
2.2
where C1 and C2 are positive constants.
3. Main Results
Theorem 3.1. Let r > 1 and 1 ≤ p < 2. Assume that {Xn , n ≥ 1} is a moving average processes
defined in 1.1, where {Yi , Fi , −∞ < i < ∞} is a martingale difference related to an increasing
sequence of σ-fields Fi and stochastically dominated by a nonnegative random variable Y . Let K be a
4
Discrete Dynamics in Nature and Society
constant. Suppose that EY rp < ∞ for rp > 1 and supi E|Yi |rp | Fi−1 ≤ K almost surely (a.s.), if
rp ≥ 2. Then for every ε > 0,
∞
nr−2 P max |Sk | > εn1/p < ∞,
3.1
1≤k≤n
n1
∞
nr−2 P
Sk sup 1/p > ε < ∞.
k
3.2
k≥n
n1
Theorem 3.2. Let the conditions of Theorem 3.1 hold. Then for every ε > 0,
∞
nr−2−1/p E max |Sk | − εn1/p
< ∞,
1≤k≤n
n1
∞
Sk r−2
n E sup 1/p − ε
< ∞.
k≥n k
n1
3.3
3.4
Remark 3.3. Let F0 ⊂ F1 ⊂ . . . be an increasing family of σ-algebras and {Xn , Fn , n ≥ 1} be a
sequence of martingale differences. Assume that for some p ≥ 2,
E |Xn |p | Fn−1 ≤ K,
3.5
a.s.,
where K is a constant not depending on n, and other conditions are satisfied, Yu 17
investigated the complete convergence of weighted sums of martingale differences. On the
other hand, under the condition
2
| Fn,k−1 ≤ K,
sup E Xn,k
a.s.,
3.6
n,k
and other conditions, Ghosal and Chandra 18 obtained the complete convergence of
martingale arrays. Thus, if rp ≥ 2, our assumption supi E|Yi |rp | Fi−1 ≤ K, a.s., is reasonable.
Chen and Li 14 obtained Theorems C and D for the case 1 ≤ rp < 2. We go on to investigate
this moving average process for the case rp > 1, especially for the case rp ≥ 2 and get the
results of 3.1–3.4. If EY rp Y log1 Y < ∞ for r > 1, 1 ≤ p < 2 and rp < 2, result
3.1 follows from Theorem C see Theorem 1.1 of Chen and Li, but we can obtain results
3.1 and 3.2 under weaker condition EY rp < ∞. On the other hand, comparing with the
conditions of Theorem D, our conditions of Theorem 3.2 are relatively simple.
4. The Proofs of Main Results
Proof of Theorem 3.1. First, we show that the moving average process 1.1 converges a.s.
under the conditions of Theorem 3.1. Since rp > 1, it has EY < ∞, following from EY rp < ∞.
On the other hand, applying Lemma 2.2 with a 1 and b 1, one has
E|Yi | ≤ 1 C2 EY IY > 1 ≤ 1 C2 EY < ∞,
−∞ < i < ∞.
4.1
Discrete Dynamics in Nature and Society
Consequently, we have by
∞
i−∞
5
|ai | < ∞ that
∞
E|ai Yin | ≤ C3
i−∞
∞
|ai | < ∞,
4.2
i−∞
which implies ∞
i−∞ ai Yin converges a.s.
Note that
Sn n
Xk k1
∞
n ∞
ai Yik k1 i−∞
ai
i−∞
in
Yk .
4.3
ki1
Let
Ynj Yj I Yj ≤ n1/p − E Yj I Yj ≤ n1/p | Fj−1 ,
−∞ < j < ∞.
4.4
Since Yj Yj I|Yj | > n1/p Ynj EYj I|Yj | ≤ n1/p | Fj−1 , we can see that
∞
nr−2 P max |Sk | > εn1/p
n1
1≤k≤n
⎞
ik
εn1/p
∞
1/p
>
⎠
≤
n P ⎝ max ai
Yj I Yj > n
2
1≤k≤n i−∞ ji1
n1
⎛
⎞
∞
∞
ik εn1/p
r−2 ⎝
1/p
⎠
n P max | Fj−1 >
ai
E Yj I Yj ≤ n
4
1≤k≤n i−∞ ji1
n1
⎞
⎛
∞
∞
ik
1/p
εn
⎠
nr−2 P ⎝ max ai
Ynj >
4
1≤k≤n i−∞ ji1
n1
∞
⎛
r−2
4.5
: H I J.
For H, by Markov’s inequality, Lemma 2.2,
∞
i−∞
|ai | < ∞ and EY rp < ∞, one has
⎞
⎛
∞
ik
∞
2
r−2 −1/p ⎝
1/p
⎠
H≤
n n
E max ai
Yj I Yj > n
ε n1
1≤k≤n i−∞ ji1
⎛
⎞
ik ∞
∞
Yj I Yj > n1/p ⎠
≤ C1 nr−2 n−1/p
|ai |E⎝ max
n1
i−∞
1≤k≤n
ji1
6
Discrete Dynamics in Nature and Society
≤ C2
∞
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
∞
m
EY Im < Y p ≤ m 1 nr−1−1/p
m1
≤ C3
∞
n1
mr−1/p EY Im < Y p ≤ m 1 ≤ C4 E|Y |rp < ∞.
m1
4.6
Meanwhile, by the martingale property, Lemma 2.2 and the proof of 4.6, it follows that
⎞
⎛
∞
ik ∞
4
r−2 −1/p ⎝
1/p
I≤
| Fj−1 ⎠
n n
E max ai
E Yj I Yj ≤ n
ε n1
1≤k≤n i−∞ ji1
⎞
⎛
∞
ik ∞
4
r−2 −1/p ⎝
1/p
| Fj−1 ⎠
n n
E max ai
E Yj I Yj > n
ε n1
1≤k≤n i−∞ ji1
≤ C1
∞
n1
≤ C2
∞
nr−2 n−1/p
∞
|ai |
i−∞
4.7
in E Yj I Yj > n1/p
ji1
nr−1−1/p E Y I Y > n1/p ≤ C3 E|Y |rp < ∞.
n1
Obviously, one can find that {Ynj , Fj−1 , −∞ < j < ∞} is a martingale difference. So, by
Markov’s inequality, Hölder’s inequality, and Lemma 2.1, we get that for any q ≥ 2,
⎧
q ⎫
⎬
q ∞
∞
ik
⎨
4
r−2 −q/p
n n
E max ai
Ynj J≤
⎩
ε n1
1≤k≤n ⎭
i−∞ ji1
⎧
⎞⎫q
⎛
⎬
q ∞
∞ ik
⎨
4
nr−2 n−q/p E
Ynj ⎠
≤
|ai |1−1/q ⎝|ai |1/q max ⎩i−∞
ε n1
1≤k≤n ⎭
ji1
⎧
q ⎫
q−1
⎬
q ∞
∞
ik
∞
⎨
4
≤
nr−2 n−q/p
Ynj |ai |
|ai |E max ⎩1≤k≤n ji1 ⎭
ε n1
i−∞
i−∞
Discrete Dynamics in Nature and Society
7
⎞q/2
⎛
in ∞
∞
2
≤ C1 nr−2 n−q/p
E Ynj
| Fj−1 ⎠
|ai |E⎝
i−∞
n1
C1
ji1
in ∞
∞
q
nr−2 n−q/p
EYnj |ai |
i−∞
n1
ji1
: C1 J1 C1 J2 .
4.8
If rp ≥ 2, then we take q large enough such that q > max{r − 1/1/p − 1/2, rp}. From
supj E|Yj |rp | Fj−1 ≤ K, a.s. and Jensen’s inequality for conditional expectation, we have
supj EYj2 | Fj−1 ≤ K 2/rp , a.s. On the other hand,
2
2
| Fj−1 E Yj2 I Yj ≤ n1/p | Fj−1 − E Yj I Yj ≤ n1/p | Fj−1
E Ynj
≤ E Yj2 I Yj ≤ n1/p | Fj−1 ,
Consequently, we obtain by
∞
i−∞
4.9
a.s., −∞ < j < ∞.
|ai | < ∞ that
⎛
⎞q/2
in ∞
∞
J1 ≤ C1 nr−2 n−q/p
E Yj2 I Yj ≤ n1/p | Fj−1 ⎠
|ai |E⎝
n1
i−∞
ji1
4.10
∞
≤ C2 nr−2−q/pq/2 < ∞,
n1
following from the fact that q > r − 1/1/p − 1/2. Meanwhile, by Cr inequality, Lemma 2.2
and ∞
i−∞ |ai | < ∞,
J2 ≤ C4
in ∞
∞
q
nr−2 n−q/p
E Yj I Yj ≤ n1/p
|ai |
n1
≤ C5
ji1
∞
∞
nr−1−q/p E Y q I Y ≤ n1/p C6 nr−1 P Y > n1/p
n1
≤ C5
i−∞
n1
∞
∞
nr−1−q/p E Y q I Y ≤ n1/p C6 nr−1−1/p E Y I Y > n1/p
n1
: C5 J21 C6 J22 .
n1
4.11
8
Discrete Dynamics in Nature and Society
Since q > rp and EY rp < ∞, one has
J21 ∞
n
nr−1−q/p E Y q I i − 11/p < Y ≤ i1/p
n1
i1
∞
E Y q I i − 11/p < Y ≤ i1/p
nr−1−q/p
∞
≤ C1
4.12
ni
i1
∞ E Y rp Y q−rp I i − 11/p < Y ≤ i1/p ir−q/p ≤ C1 EY rp < ∞.
i1
By the proof of 4.6,
J22 ∞
nr−1−1/p E Y I Y > n1/p ≤ CEY rp < ∞.
4.13
n1
If rp < 2, then we take q 2. Similar to the proofs of 4.8, and 4.11, it has
J ≤ C1
in
∞
∞
2
nr−2 n−2/p
EYnj
|ai |
n1
i−∞
ji1
in ∞
∞
≤ C2 nr−2 n−2/p
E Yj2 I Yj ≤ n1/p ≤ C3 EY rp < ∞,
|ai |
n1
i−∞
4.14
ji1
following from q > rp, 4.12, and 4.13. Therefore, 3.1 follows from 4.5–4.13 and the
inequality above.
Inspired by the proof of Theorem 12.1 of Gut 24, it can be checked that
∞
n1
r−2
n
P
m
∞ 2
−1
Sk 2/p
r−2
sup 1/p > 2 ε n P sup
k
k≥n
m1 n2m−1
≤2
2−r
∞
P
≤2
∞
m
2
−1
Sk 2/p
sup 1/p > 2 ε
2mr−2
k
m−1
k≥2
m1
2−r
k≥n
2
∞
2
P
Sk 2/p
sup 1/p > 2 ε
k
m−1
mr−1
2
m1
n2m−1
mr−1
m1
2−r
Sk 2/p
k1/p > 2 ε
P
k≥2
Sk 2/p
sup max 1/p > 2 ε
2l−1 ≤k<2l k
l≥m
Discrete Dynamics in Nature and Society
9
≤ 22−r
∞
2mr−1
m1
2
2−r
∞ P max |Sk | > ε2l1/p
1≤k≤2l
lm
l
∞ l1/p
P max |Sk | > ε2
2mr−1
1≤k≤2l
l1
m1
∞
lr−1
l1/p
≤ C1 2
P max |Sk | > ε2
1≤k≤2l
l1
: D.
4.15
If r < 2, then
D 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
≤ 22−r C1
∞
4.16
1≤k≤n
l1 n2l
nr−2 P max |Sk | > εn1/p .
1≤k≤n
n1
Otherwise,
D C1
l1
−1
∞ 2
2lr−2 P max |Sk | > ε2l1/p
1≤k≤2l
l1 n2l
≤ C1
l1
−1
∞ 2
nr−2 P max |Sk | > εn1/p
1≤k≤n
l1 n2l
≤ C1
∞
nr−2 P max |Sk | > εn1/p .
1≤k≤n
n1
Combining 3.1 with these inequalities above, we obtain 3.2 immediately.
Proof of Theorem 3.2. For all ε > 0, it has
∞ ∞
∞
nr−2−1/p E max |Sk | − εn1/p
nr−2−1/p
P max |Sk | − εn1/p > t dt
n1
1≤k≤n
n1
1≤k≤n
0
n1/p ∞
nr−2−1/p
P max |Sk | − εn1/p > t dt
n1
∞
n1
1≤k≤n
0
nr−2−1/p
∞
n1/p
P max |Sk | − εn1/p > t dt
1≤k≤n
4.17
10
Discrete Dynamics in Nature and Society
≤
∞
nr−2 P max |Sk | > εn1/p
1≤k≤n
n1
∞
nr−2−1/p
∞
n1/p
n1
P max |Sk | > t dt.
1≤k≤n
4.18
By Theorem 3.1, in order to proof 3.3, we only have to show that
∞ ∞
nr−2−1/p
P max |Sk | > t dt < ∞.
n1/p
n1
1≤k≤n
4.19
For t > 0, denote
Ytj Yj I Yj ≤ t − E Yj I Yj ≤ t | Fj−1 ,
−∞ < j < ∞.
4.20
Since Yj Yj I|Yj | > t Ytj EYj I|Yj | ≤ t | Fj−1 , it is easy to see that
∞ ∞
nr−2−1/p
P max |Sk | > t dt
n1
1≤k≤n
n1/p
⎞
⎛
∞
∞
ik
∞
t
r−2−1/p
n
P ⎝ max ai
Yj I Yj > t > ⎠dt
≤
1≤k≤n 2
n1/p
i−∞ ji1
n1
⎛
⎞
∞
∞
∞
ik
t
r−2−1/p
n
P ⎝ max ai
Ytj > ⎠dt
1≤k≤n 4
n1/p
i−∞ ji1
n1
⎛
⎞
∞
∞
∞
ik
t
nr−2−1/p
P ⎝ max ai
E Yj I Yj ≤ t | Fj−1 > ⎠dt
1≤k≤n 4
n1/p
i−∞ ji1
n1
: I1 I2 I3 .
4.21
By Markov’s inequality, Lemma 2.2 and EY rp < ∞,
⎞
⎛
∞
∞
ik
∞
r−2−1/p
−1 ⎝
I1 ≤ 2 n
t E max ai
Yj I Yj > t ⎠dt
1≤k≤n n1/p
i−∞ ji1
n1
⎞
⎛
∞
∞
ik ∞
r−2−1/p
−1
Yj I Yj > t ⎠dt
≤2 n
t
|ai |E⎝ max
n1/p
n1
≤ C1
∞
n1
nr−1−1/p
∞
n1/p
i−∞
1≤k≤n
ji1
t−1 EY IY > tdt
Discrete Dynamics in Nature and Society
C1
∞
∞
n
m1/p
mn
nr−1−1/p
∞
∞
t−1 E Y I Y > m1/p dt
m1/p−1−1/p E Y I Y > m1/p
mn
n1
C2
∞ m1
1/p
r−1−1/p
n1
≤ C2
11
m
m−1 E Y I Y > m1/p
nr−1−1/p
m1
≤ C3
∞
n1
mr−1−1/p E Y I Y > m1/p ≤ C4 EY rp < ∞.
m1
4.22
Since {Ytj , Fj−1 , −∞ < j < ∞} is a martingale difference, we have by Markov’s inequality,
Hölder’s inequality, and Lemma 2.1 that for any q ≥ 2,
I2 ≤ 4q
∞
nr−2−1/p
q ⎞
∞
ik
t−q E⎝ max ai
Ytj ⎠dt
1≤k≤n n1/p
i−∞ ji1
⎛
∞
n1
⎧
⎞⎫q
⎛
∞
⎬
∞
∞ ik
⎨
≤ C nr−2−1/p
t−q E
Ytj ⎠ dt
|ai |1−1/q ⎝|ai |1/q max ⎩i−∞
1≤k≤n ⎭
n1/p
n1
ji1
∞
∞
≤ C nr−2−1/p
t−q
n1/p
n1
≤ C1
∞
r−2−1/p
n
n1/p
∞
r−2−1/p
n
n1
∞
q−1
|ai |
i−∞
∞
n1
C1
t
−q
∞
n1/p
∞
⎧
⎨
4.23
⎛
⎞q/2
in 2
E Ytj | Fj−1 ⎠ dt
|ai |E⎝
i−∞
t−q
q ⎫
⎬
ik
Ytj dt
|ai |E max ⎩1≤k≤n ji1 ⎭
i−∞
∞
∞
ji1
|ai |
i−∞
in q
EYtj dt
ji1
: C1 I21 C1 I22 .
If rp ≥ 2, then we take large enough q such that q > max{r − 1/1/p − 1/2, rp}. By
supj E|Yj |rp | Fj−1 ≤ K, a.s. and Jensen’s inequality for conditional expectation, it has
supj EYj2 | Fj−1 ≤ K 2/rp , a.s.. Meanwhile,
2
2
E Ynj
| Fj−1 E Yj2 I Yj ≤ n1/p | Fj−1 − E Yj I Yj ≤ n1/p | Fj−1
≤ E Yj2 I Yj ≤ n1/p | Fj−1 ,
a.s., −∞ < j < ∞.
4.24
12
Discrete Dynamics in Nature and Society
Thus, by
I21
∞
i−∞
|ai | < ∞, one has that
⎛
⎞q/2
∞
in
∞
∞
C1 nr−2−1/p
t−q
Znj ⎠ dt
|ai |E⎝
n1/p
n1
i−∞
ji1
⎞q/2
⎛
∞
in
∞
∞
r−2−1/p
−q
2
1/p
| Fj−1 ⎠ dt
≤ C1 n
t
2E Yj I Yj ≤ n
|ai |E⎝
n1/p
n1
≤ C2
ji1
4.25
∞
∞
∞
nr−2−1/pq/2
t−q dt ≤ C3 nr−2−1/pq/2 · n−q1/p
n1/p
n1
C3
i−∞
n1
∞
nr−2q/2−q/p < ∞,
n1
following from the fact that q > r − 1/1/p − 1/2. We also have by Cr inequality and
Lemma 2.2 that
I22 ≤ C2
∞
in
∞
∞
q nr−2−1/p
t−q
E Yj I Yj ≤ t dt
|ai |
n1/p
n1
i−∞
∞ m1
∞
r−1−1/p
≤ C3 n
C4
m1/p
mn
n1
∞
nr−1−1/p
∞
n1/p
n1
ji1
1/p
t−q EY q IY ≤ tdt
P Y > tdt
∗
∗∗
C3 I22
.
: C2 I22
Since q > rp and EY rp < ∞, it follows that
∗
I22
≤ C1
∞
nr−1−1/p
∞
m1/p−1−q/p E Y q I Y ≤ m 11/p
mn
n1
C1
∞
m
m1/p−1−q/p E Y q I Y ≤ m 11/p
nr−1−1/p
m1
≤ C2
∞
n1
mr−1−q/p E Y q I Y ≤ m 11/p
m1
C2
∞
mr−1−q/p
m1
E Y q I i − 11/p < Y ≤ i1/p
m1
i1
4.26
Discrete Dynamics in Nature and Society
C2
∞
13
mr−1−q/p E Y rp Y q−rp I m1/p < Y ≤ m 11/p
m1
C2
∞
mr−1−q/p
m1
≤ 2q−rp/p C2
m E Y q I i − 11/p < Y ≤ i1/p
i1
∞
m−1 E Y rp I m1/p < Y ≤ m 11/p
m1
C2
∞
∞
E Y q I i − 11/p < Y ≤ i1/p
mr−1−q/p
mi
i1
≤ 2q−rp/p C2
∞
m−1 E Y rp I m1/p < Y ≤ m 11/p
m1
C2
∞
E Y rp Y q−rp I i − 11/p < Y ≤ i1/p ir−q/p
i1
≤ C3 EY rp < ∞.
4.27
From the proof of 4.22,
∗∗
I22
≤
∞
∞
nr−1−1/p
t−1 EY IY > tdt ≤ CEY rp < ∞.
n1/p
n1
4.28
If rp < 2, then we take q 2. Similar to the proofs of 4.23 and 4.26, we get that
I2 ≤ C1
∞
r−2−1/p
∞
n
n1
n1/p
t−2
∞
|ai |
i−∞
in
EYtj2 dt ≤ C2 EY rp < ∞,
4.29
ji1
following from q > rp, 4.27 and 4.28. Consequently, by 4.18–4.28, Theorem 3.1 and
inequality above, 3.3 holds true.
Now, we turn to prove 3.4. Similar to the proof of 3.2, we have that
∞
n1
r−2
n
Sk 2/p
E sup 1/p − ε2
k
k≥n
∞ Sk 2/p
n
P sup 1/p > ε2 t dt
0
k≥n k
n1
∞ m
∞ 2
−1
Sk r−2
2/p
n
P sup 1/p > ε2 t dt
0
k≥n k
m1 n2m−1
∞
r−2
14
Discrete Dynamics in Nature and Society
m
2
−1
∞ ∞
Sk 2−r
2/p
≤2
P sup 1/p > ε2 t dt
2mr−2
m−1
k≥2m−1 k
m1 0
n2
∞
∞
Sk 2−r
mr−1
2/p
≤2
2
P sup 1/p > ε2 t dt
0
k≥2m−1 k
m1
∞ ∞
Sk 2−r
mr−1
2/p
2
2
P sup max 1/p > ε2 t dt
l−1
l
0
l≥m 2 ≤k<2 k
m1
∞
∞ ∞ 2−r
mr−1
2/p
l−1/p
2
P max |Sk | > ε2 t 2
dt
≤2
m1
22−r
l
∞ ∞ P max |Sk | > ε22/p t 2l−1/p dt 2mr−1
l1
≤ 22−r
∞
1≤k≤2l
0
m1
∞ 2lr−1
P max |Sk | > ε22/p t 2l−1/p dt
1≤k≤2l
0
l1
≤ C1
1≤k≤2l
0
lm
let s 2l−1/p t
∞ ∞
2lr−1−1/p
P max |Sk | > ε2l1/p s ds : F.
1≤k≤2l
0
l1
4.30
If r < 2 1/p, then
F 221/p−r C1
∞ l1
−1
∞ 2
2l1r−2−1/p
P max |Sk | > ε2l1/p s ds
0
l1 n2l
≤ 221/p−r C1
∞ l1
−1
∞ 2
nr−2−1/p
P max |Sk | > εn1/p s ds
≤2
C1
∞
r−2−1/p
n
n1
4.31
1≤k≤n
0
l1 n2l
21/p−r
1≤k≤2l
E max |Sk | − εn
1/p
1≤k≤n
< ∞.
Otherwise,
F C1
∞ l1
−1
∞ 2
2lr−2−1/p
P max |Sk | > ε2l1/p s ds
0
l1 n2l
≤ C1
∞ l1
−1
∞ 2
nr−2−1/p
P max |Sk | > εn1/p s ds
l1 n2l
≤ C1
1≤k≤2l
0
1≤k≤n
∞
nr−2−1/p E max |Sk | − εn1/p
< ∞.
n1
1≤k≤n
Therefore, 3.4 holds true following from 3.3.
4.32
Discrete Dynamics in Nature and Society
15
Acknowledgments
The authors are grateful to Editor Chuanxi Qian and an anonymous referee for their
careful reading and insightful comments. This work is supported by the National Natural
Science Foundation of China 11171001, 11126176, HSSPF of the Ministry of Education of
China 10YJA910005, Natural Science Foundation of Anhui Province 1208085QA03, and
Provincial Natural Science Research Project of Anhui Colleges KJ2010A005.
References
1 I. A. Ibragimov, “Some limit theorems for stationary processes,” Theory of Probability and Its
Applications, vol. 7, no. 4, pp. 361–392, 1962.
2 R. M. Burton and H. Dehling, “Large deviations for some weakly dependent random processes,”
Statistics & Probability Letters, vol. 9, no. 5, pp. 397–401, 1990.
3 D. L. Li, M. B. Rao, and X. C. Wang, “Complete convergence of moving average processes,” Statistics
& Probability Letters, vol. 14, no. 2, pp. 111–114, 1992.
4 L. X. Zhang, “Complete convergence of moving average processes under dependence assumptions,”
Statistics & Probability Letters, vol. 30, no. 2, pp. 165–170, 1996.
5 Y. X. Li and L. X. Zhang, “Complete moment convergence of moving-average processes under
dependence assumptions,” Statistics & Probability Letters, vol. 70, no. 3, pp. 191–197, 2004.
6 T.-S. Kim, M.-H. Ko, and Y.-K. Choi, “Complete moment convergence of moving average processes
with dependent innovations,” Journal of the Korean Mathematical Society, vol. 45, no. 2, pp. 355–365,
2008.
7 P. Y. Chen, T.-C. Hu, and A. Volodin, “Limiting behaviour of moving average processes under φmixing assumption,” Statistics & Probability Letters, vol. 79, no. 1, pp. 105–111, 2009.
8 X. C. Zhou, “Complete moment convergence of moving average processes under φ-mixing
assumptions,” Statistics & Probability Letters, vol. 80, no. 5-6, pp. 285–292, 2010.
9 W. Z. Yang, X. J. Wang, N. X. Ling, and S. H. Hu, “On complete convergence of moving average
process for AANA sequence,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 863931, 24
pages, 2012.
10 Y. S. Chow, “On the rate of moment convergence of sample sums and extremes,” Bulletin of the Institute
of Mathematics, vol. 16, no. 3, pp. 177–201, 1988.
11 P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,” Proceedings of the
National Academy of Sciences of the United States of America, vol. 33, pp. 25–31, 1947.
12 Q. M. Shao, “A moment inequality and its applications,” Acta Mathematica Sinica, vol. 31, no. 6, pp.
736–747, 1988.
13 S. H. Sung, “Complete convergence for weighted sums of ρ∗ -mixing random variables,” Discrete
Dynamics in Nature and Society, Article ID 630608, 13 pages, 2010.
14 P. Y. Chen and Y. M. Li, “Limiting behavior of moving average processes of martingale difference
sequences,” Acta Mathematica Scientia A, vol. 31, no. 1, pp. 179–187, 2011.
15 Y. S. Chow, “Convergence of sums of squares of martingale differences,” Annals of Mathematical
Statistics, vol. 39, pp. 123–133, 1968.
16 P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, NY,
USA, 1980.
17 K. F. Yu, “Complete convergence of weighted sums of martingale differences,” Journal of Theoretical
Probability, vol. 3, no. 2, pp. 339–347, 1990.
18 S. Ghosal and T. K. Chandra, “Complete convergence of martingale arrays,” Journal of Theoretical
Probability, vol. 11, no. 3, pp. 621–631, 1998.
19 S. H. Hu, “Fixed-design semiparametric regression for linear time series,” Acta Mathematica Scientia
B, vol. 26, no. 1, pp. 74–82, 2006.
20 S. H. Hu, X. Li, W. Yang, and X. Wang, “Maximal inequalities for some dependent sequences and their
applications,” Journal of the Korean Statistical Society, vol. 40, no. 1, pp. 11–19, 2011.
21 S. H. Hu, C. H. Zhu, Y. B. Chen, and L. Wang, “Fixed-design regression for linear time series,” Acta
Mathematica Scientia, vol. 22, no. 1, pp. 9–18, 2002.
22 L. Tran, G. Roussas, S. Yakowitz, and B. Truong Van, “Fixed-design regression for linear time series,”
The Annals of Statistics, vol. 24, no. 3, pp. 975–991, 1996.
16
Discrete Dynamics in Nature and Society
23 Q. Y. Wu, Probability Limit Theory for Mixed Sequence, Science Press, Beijing, China, 2006.
24 A. Gut, Probability: A Graduate Course, Springer ScienceBusiness Media,, New York, NY, USA, 2005.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014