Research Article On Complete Convergence of Moving Average Process for AANA Sequence

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1 Discrete Dynamics in Nature and Society Volume 2012, Article ID , 24 pages doi: /2012/ Research Article On Complete Convergence of Moving Average Process for AANA Sequence Wenzhi Yang, 1 Xuejun Wang, 1 Nengxiang Ling, 2 and Shuhe Hu 1 1 School of Mathematical Science, Anhui University, Hefei , China 2 School of Mathematics, Hefei University of Technology, Hefei , 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 wor is properly cited. We investigate the moving average process such that X n i1 Y in, n 1, where i1 < and {Y i, 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 {Y i, <i< } is a doubly infinite sequence of identically distributed random variables with E Y 1 <.Let{, <i< } be an absolutely summable sequence of real numbers and X n Y in, n i be the moving average process based on the sequence {Y i, < i < }. As usual, S n n X,n 1 denotes the sequence of partial sums. Under the assumption that {Y i, <i< } is a sequence of independent identically distributed random variables, various results of the moving average process {X n,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 {X n,n 1}.

2 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 {Y i, < i < } is a sequence of identically distributed ϕ-mixing random variables with ϕ1/2 m < and {X n,n 1} is as in 1.1.Lethx > 0 x >0 be a slowly varying function and 1 p<2, r 1. IfY 1 0 and E Y 1 rp h Y 1 p <,then n r 2 hnp S n εn 1/p <, ε > Theorem B. Suppose that {Y i, < i < } is a sequence of identically distributed ϕ-mixing random variables with EY 1 0, EY 2 1 < and ϕ1/2 m < and {X n, n 1} is as in 1.1. Let hx > 0 x >0 be a slowly varying function and 1 p<2, r>1.ife Y 1 rp h Y 1 p <,then where x max{x, 0}. n r 2 1/p hne S n εn 1/p <, ε >0, 1.3 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<2and p>0. Assume that {X n,n 1} is a mean zero sequence of identically distributed ρ-mixing random variables with the maximal correlation coefficient rate ρ2/s 2 n <,wheres 2 if p<2and s>pif p 2. Denote S n n i1 X i,n 1. Then E X 1 r <, if p < r, E X 1 r log1 X 1 ] <, if p r, E X 1 p <, if p > r, E sup X n 1.4 p <, n 1 n 1 n 1/r E sup S n p < n 1/r 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,Chow11,Shao12,Lietal.3, Zhang 4,LiandZhang5, Chen and Gan 7,Kimetal.6, Sung13 15, Chen and Li 16, Zhou and Lin 17, and so forth. Inspired by Zhang 4, Kimetal.6, Chen and Gan 7, Sung13 15, and other papers above, we investigate the limit behavior of moving average process under AANA

3 Discrete Dynamics in Nature and Society 3 sequence, which is weaer 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 {X n,n 1} is stochastically dominated by a nonnegative random variable X if sup P X n >t CPX >t for some positive constant C and t n 1 Definition 1.1. A finite collection of random variables X 1,X 2,...,X n is said to be negatively associated NA if for every pair of disjoint subsets A 1,A 2 of {1, 2,...,n}, Cov { fx i : i A 1, g X j : j A 2 } 0, 1.6 whenever f and g are coordinatewise nondecreasing such that this covariance exists. An infinite sequence {X n,n 1} is NA if every finite subcollection is NA. Definition 1.2. A sequence {X n, n 1} of random variables is called asymptotically almost negatively associated AANA if there exists a nonnegative sequence qn 0asn such that Cov fx n,gx,x n2,...,x n qn Var fx n Var gx,x n2,...,x n ] 1/2, 1.7 for all n, 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, Wangetal.22, Koetal. 23, Yuan and An 24, and Wang et al. 25, 26 among others. For simplicity, in this paper we consider the moving average process: X n Y in, n 1, 1.8 i1 where i1 < and {Y i, 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 {X n,n 1} be a sequence of AANA random variables with mixing coefficients {qn, n 1}. Iff 1,f 2,... are all nondecreasing or nonincreasing continuous functions, then {f n X n,n 1} is still a sequence of AANA random variables with mixing coefficients {qn, n 1}.

4 4 Discrete Dynamics in Nature and Society Remar 1.4. Lemma 1.3 comes from Lemma 2.1 of Yuan and An 24, but the functions of f 1,f 2,... in Lemma 2.1 of Yuan and An 24 are written to be all nondecreasing or nonincreasing functions. According to the definition of AANA, f 1,f 2,... should be all nondecreasing or nonincreasing continuous functions. Lemma 1.5 cf. Wang et al. 25, Lemma 1.4. Let 1 <p 2and {X n,n 1} be a mean zero sequence of AANA random variables with mixing coefficients {qn, n 1}.If q2 n <,then there exists a positive constant C p depending only on p such that p n E max X i C p E X i p, i1 i1 1.9 for all n 1, wherec p 2 p 2 2 p 6p p q2 n p 1. Lemma 1.6 cf. Wu 27, Lemma Let {X n,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 X n α I X n b ] C 1 {EX α IX b b α PX >b}, E X n α I X n >b ] C 2 EX α IX >b, 1.10 where C 1 and C 2 are positive constants. Throughout the paper, IA is the indicator function of set A, x max{x, 0} and C, C 1,C 2,... 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 {X n,n 1} is a moving average process defined in 1.8, where{y i, 1 i< } is a mean zero sequence of AANA random variables with q2 n < and stochastically dominated by a nonnegative random variable Y.IfEY rp <,then for every ε>0, n r 2 P max S >εn 1/p <, 2.1 n r 2 P sup S >ε <. 2.2 n 1/p

5 Discrete Dynamics in Nature and Society 5 Theorem 2.2. Let the conditions of Theorem 2.1 hold. Then for every ε>0, n r 2 1/p E max S εn 1/p <, 2.3 n r 2 E sup S ε <. n 1/p 2.4 Theorem 2.3. Let 0 <r<2 and 0 <p<2. Assume that {X n,n 1} is a moving average process defined in 1.8, where{y i, 1 i< } is a mean zero sequence of AANA random variables with q2 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 <, if r > 1, E Y log1 Y ] <, if 0 <r<1, ] E Ylog 2 1 Y <, if r 1, E Y r log1 Y ] <, if r > 1, E Y log1 Y ] <, if p 1, EY p <, ifp > Then E sup S n n 1 n 1/r p < 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 1andb 1, one has E Y i 1 C 2 EYIY >1 1 C 2 EY <, 1 i<. 3.1 Consequently, by the condition i1 <, we have that E Y in C 3 <, 3.2 i1 i1 which implies i1 Y in converges almost surely.

6 6 Discrete Dynamics in Nature and Society Note that n n X Y i i1 in i1 i1 Y, n Since r>1andey rp <, one has EY p <. Combining EY p < with EY j 0, i1 < and Lemma 1.6, we can find that n 1/p max i i1 ji1 E n 1/p max in n 1/p i1 ] Y j I Yj n 1/p i i1 ji1 ji1 E ] Y j I Yj >n 1/p ] E Yj I Yj >n 1/p CE n 1/p p 1 YI Y>n 1/p] CE Y p I Y>n 1/p] 0 as n. 3.4 Meanwhile, n 1/p max i i1 ji1 in n 1/p i1 n 1/p max i i1 ji1 n 1/p E I ji1 in n 1/p i1 Y j < n 1/p] E ] Yj I Yj >n 1/p CE n 1/p E ji1 I Y j >n 1/p] E Y j I Y j >n 1/p] CE Y p I Y p I Y>n 1/p] 0 as n, Y>n 1/p] 0 as n. 3.5 Let Y nj n 1/p I Y j < n 1/p Y j I Yj n 1/p n 1/p I Y j >n 1/p, j 1, Ỹ nj Y nj EY nj, j

7 Discrete Dynamics in Nature and Society 7 Hence, for for all ε>0, there exists an n 0 such that n 1/p max i i1 ji1 EY nj < ε 4, n n Denote Y nj n1/p I Y j < n 1/p n 1/p I Y j >n 1/p Y j I Yj >n 1/p, j Noting that Y j Y nj Y nj, j 1, we can find n r 2 P max S >εn 1/p n r 2 P max n r 2 P max n r 2 P max n r 2 P max i Y nj i1 ji1 i i1 ji1 i Y nj i1 ji1 i i1 ji1 C n r 2 P max nn 0 C n r 2 P max n r 2 P max > εn1/p 2 Y nj > εn1/p 2 > εn1/p 2 Ỹ nj > εn1/p 4 i i1 ji1 i Y nj i1 ji1 i i1 ji1 EY nj > εn1/p 4 > εn1/p 2 Ỹ nj > εn1/p : C I J.

8 8 Discrete Dynamics in Nature and Society For I, by Marov s inequality, i1 <, Y nj Y j I Y j >n 1/p, Lemma 1.6 and EY rp <,ithas I 2 n r 2 n 1/p E max ε i Y nj i1 ji1 i C 1 n r 2 n 1/p E max Yj I Yj >n 1/p i1 ji1 C 2 n r 1 1/p Y >n 1/p] C 2 n r 1 1/p EYIm <Y p m 1 mn m C 2 EYIm <Y p m 1 n r 1 1/p C 3 m r 1/p EYIm <Y p m 1 CEY rp < Since f j x n 1/p Ix < n 1/p xi x n 1/p n 1/p Ix >n 1/p is a nondecreasing continuous function of x, we can find by using Lemma 1.3 that {Ỹ nj, 1 j< } is a mean zero AANA sequence and EỸ 2 nj EY 2 nj, Y 2 nj Y 2 j I Y j n 1/p n 2/p I Y j >n 1/p, j 1. Consequently, by the property of AANA, Marov s inequality, Hölder s inequality, Lemma 1.5, C r inequality, and Lemma 1.6, we can chec that 4 2 J n r 2 n 2/p E ε max i i1 ji1 2 Ỹ nj 4 2 n r 2 n 2/p E a ε i 1/2 ai 1/2 i max Ỹ nj ε n r 2 n 2/p sup E i1 i 1 max in C 1 n r 2 n 2/p sup i 1 i 1 i1 EỸ 2 nj ji1 ji1 i ji1 2 Ỹ nj ji1 in { C 2 n r 2 n 2/p sup E Y 2 j I Y j n 1/p] n 2/p E I Y j >n 1/p]} 2

9 Discrete Dynamics in Nature and Society 9 C 3 n r 1 2/p E Y 2 I Y n 1/p] C 4 n r 1 P Y >n 1/p C 3 n r 1 2/p E Y 2 I Y n 1/p] C 4 n r 1 1/p Y >n 1/p] : C 3 J 1 C 4 J Since rp < 2, it can be seen by EY rp < that J 1 n n r 1 2/p E Y 2 I i 1 1/p <Y i 1/p] i1 E Y 2 I i 1 1/p <Y i 1/p] i1 i1 n r 1 2/p ni C 1 E Y rp Y 2 rp I i 1 1/p <Y i 1/p] i r 2/p C 2 EY rp < By the proof of 3.10, we have J 2 CEY rp <. Therefore, 2.1 follows from 3.9, 3.10, 3.11, and3.12. Inspired by the proof of Theorem 12.1 of Gut 28, it can be checed that n r 2 P sup S > 22/p ε n 1/p 2 m 1 n r 2 P sup S n2 m 1 n 1/p > 22/p ε 2 2 r P sup S 2 > m 1 22/p ε 2 m 1 1/p 2 m 1 1/p n2 m 1 2 mr r 2 mr 1 P sup S > 22/p ε 2 2 r 2 mr 1 P sup max S > 22/p ε lm l m 2 l 1 <2 l 1/p 2 2 r 2 mr 1 P max S >ε2 l1/p 1 2 l l 2 2 r P max S >ε2 l1/p 1 2 l l1 2 mr 1

10 10 Discrete Dynamics in Nature and Society C 1 2 lr 1 P max S >ε2 l1/p 1 2 l l1 2 2 r C 1 l1 2 2 r C 1 l1 2 l1 1 n2 l 2 l1r 2 P 2 l1 1 n2 l n r 2 P max S >ε2 l1/p 1 2 l max S >εn 1/p 2 2 r C 1 n r 2 P max S >εn 1/p. since r< Combining 2.1 with the inequality above, we obtain 2.2 immediately. Proof of Theorem 2.2. For all ε>0, it has n r 2 1/p E max S εn 1/p n r 2 1/p 0 n 1/p n r 2 1/p 0 P max S εn 1/p >t dt P max S εn 1/p >t dt n r 2 1/p P max S εn 1/p >t dt n 1/p n r 2 P max S >εn 1/p n r 2 1/p P n 1/p max S >t dt By Theorem 2.1, in order to prove 2.3, we only have to prove that n r 2 1/p P max S >t dt <. n 1/p 3.15 For t>0, let Y tj ti Y j < t Y j I Yj t ti Yj >t, j 1, Ỹ tj Y tj EY tj, j 1, 3.16 Y tj ti Y j < t ti Y j >t Y j I Yj >t, j 1.

11 Discrete Dynamics in Nature and Society 11 Since Y j Y tj Y tj, j 1, it is easy to see that n r 2 1/p P max S >t dt n 1/p n r 2 1/p P max n 1/p n r 2 1/p P max n 1/p n r 2 1/p P max n 1/p i Y tj i1 ji1 i i1 i i1 > t dt 2 tj > ji1ỹ t dt 4 tj > ji1ey t dt : I 1 I 2 I 3. For I 1, by Marov s inequality, Y tj Y j I Y j >t, Lemma 1.6 and EY rp <,wegetthat I 1 2 n r 2 1/p t 1 E max n 1/p i Y tj i1 ji1 dt i 2 n r 2 1/p t 1 E max Y j I Y j >t dt n 1/p i1 ji1 C 1 n r 1 1/p t 1 EYIY >tdt n 1/p C 1 n r 1 1/p mn mn 1/p m 1/p t 1 Y>m 1/p] dt C 2 n r 1 1/p m 1/p 1 1/p Y >m 1/p] C 2 m 1 Y >m 1/p] m C 3 m r 1 1/p E YI n r 1 1/p Y>m 1/p] C 4 EY rp <. 3.18

12 12 Discrete Dynamics in Nature and Society From the fact that {Ỹ tj, 1 j< } is a mean zero AANA sequence and EỸ 2 tj EY tj, Y 2 tj Y 2 j I Y j tt 2 I Y j >t, j 1, similar to the proof of 3.11, we have I n r 2 1/p t 2 E max n 1/p 4 2 n r 2 1/p n 1/p t 2 i1 i i1 ji1 Ỹ 2 tj dt 2 sup E i 1 max 2 Ỹ tj dt i ji1 in C 1 n r 2 1/p t 2 sup EỸ 2 n 1/p i 1 ji1 tj dt 3.19 in { C 2 n r 2 1/p t 2 sup E Y 2 j I ] Yj t t 2 E I ] } Yj >t dt n 1/p i 1 ji1 C 3 n r 1 1/p t 2 E Y 2 IY t ]dt C 4 n r 1 1/p PY >tdt n 1/p n 1/p : C 3 I 21 C 4 I 22. It follows from rp < 2andEY rp < that I 21 n r 1 1/p mn 1/p mn m 1/p ] t 2 E Y 2 IY t dt C 1 n r 1 1/p m 1/p 1 2/p E Y 2 I Y m 1 1/p] C 1 m 1/p 1 E Y 2 I Y m 1 1/p] m C 2 m r 1 2/p E Y 2 I Y m 1 1/p] C 2 m r 1 2/p i1 n r 1 1/p E Y 2 I i 1 1/p <Y i 1/p] C 2 m r 1 2/p E Y 2 I m 1/p <Y m 1 1/p] m C 2 m r 1 2/p E Y 2 I i 1 1/p <Y i 1/p] C 2 i1 m r 1 2/p E Y rp Y 2 rp I m 1/p <Y m 1 1/p]

13 Discrete Dynamics in Nature and Society 13 C 2 E Y 2 I i 1 1/p <Y i 1/p] m r 1 2/p i1 mi 2 2 rp/p C 2 m 1 E Y rp I m 1/p <Y m 1 1/p] C 3 E Y rp Y 2 rp I i 1 1/p <Y i 1/p] i r 2/p C 4 EY rp <. i By the proof of 3.18, one has I 22 n r 1 1/p t 1 EYIY >tdt CEY rp <. n 1/p 3.21 On the other hand, by the property EY j 0, we have i max tj i1 ji1ey i max E Yj I i1 ji1{ Y j t ] te I Y j < t ] te I Y j >t ]} i max E Yj I i1 ji1{ Y j >t ] te I Y j < t ] te I Y j >t ]} in 2 E ] Yj I Yj >t. i1 ji Thus, by Lemma 1.6 and the proof of 3.18, it can be seen that I 3 4 n r 2 1/p t 1 max n 1/p i i1 ji1 EY tj dt in 8 n r 2 1/p t 1 E ] Yj I Yj >t dt n 1/p i1 ji1 C 1 n r 1 1/p t 1 EYIY >tdt C 2 EY rp <. n 1/p 3.23 Consequently, by 3.14, 3.17, 3.18, 3.19, 3.20, 3.21,and3.23 and Theorem 2.1, 2.3 holds true.

14 14 Discrete Dynamics in Nature and Society Next, we prove 2.4. Itiseasytoseethat n r 2 E sup S ε22/p n n r 2 0 1/p 2 m 1 n r 2 n2 m r 2 2 r 2 2 r 0 2 mr mr 1 0 P sup S >ε22/p t dt n 2 2 r 2 mr 1 1/p P sup S >ε22/p t dt 2 m 1 n 1/p 1/p P sup S >ε22/p t dt lm 2 m 1 n2 m 1 2 mr 2 P sup S 2 1/p >ε22/p t dt m 1 P sup max S >ε22/p t dt 0 l m 2 l 1 <2 l 1/p P max S > ε2 2/p t 2 l 1/p dt 1 2 l l 2 2 r P max S > ε2 2/p t 2 l 1/p dt l l 2 2 r 2 lr 1 P max S > ε2 2/p t 2 l 1/p dt l l C 1 2 lr 1 1/p P max S >ε2 l1/p s ds l l 2 21/p r C /p r C 1 2 l1 1 2 l1r 2 1/p l1 n2 l 0 2 l1 1 n r 2 1/p l1 n2 l 0 2 mr 1 let s 2 l 1/p t P max S >ε2 l1/p s ds 1 2 l P max S >εn 1/p s ds 2 21/p r C 1 n r 2 1/p E max S εn 1/p <. since r< Therefore, 2.4 holds true following from 2.3.

15 Discrete Dynamics in Nature and Society 15 Proof of Theorem 2.3. Similar to the proof of Theorem 2.1, by i1 < and EY <, i1 Y in converges almost surely. It can be seen that E sup S n p n 1 n 1/r 0 P sup S n dt >t1/p n 1 n 1/r 2 p/r P sup S n 2 p/r n 1 n 1/r dt >t1/p 2 p/r P sup max S n 2 p/r n<2 n 1/r dt >t1/p 2 p/r P max S n 2 p/r 2 1 n<2 n 1/r dt >t1/p 2 p/r 2 p/r P max S n > 2 1/r t 1/p dt 1 n 2 2 p/r 2 p/r 2 p/r P max S n >s 1/p ds. 2 p/r 1 n 2 let s 2 1p/r t 3.25 For s 1/p > 0, let Y sj s 1/p I Y j < s 1/p Y j I Yj s 1/p s 1/p I Y j >s 1/p, j 1, Ỹ sj Y sj EY sj, j 1, Y sj s1/p I Y j < s 1/p s 1/p I Y j >s 1/p Y j I Yj >s 1/p, j Since Y j Y sj Y sj, j 1, then 2 p/r P max S n >s 1/p ds 2 p/r 1 n 2 2 p/r P max 2 p/r 1 n 2 2 p/r P max 2 p/r 1 n 2 2 p/r P max 2 p/r 1 n 2 in Y sj i1 ji1 in i1 ji1 in i1 ji1 > s1/p ds 2 Ỹ sj > s1/p ds 4 EY sj > s1/p ds : H 1 H 2 H 3.

16 16 Discrete Dynamics in Nature and Society For H 1, similar to 3.18, by Marov s inequality and Lemma 1.6, one has H p/r s 1/p E max 2 p/r 1 n 2 in Y sj i1 ji1 ds in 2 2 p/r s 1/p E max Yj I Yj >s 1/p ds 2 p/r i1 1 n 2 ji1 C 1 2 p/r s 1/p Y>s 1/p] ds 2 p/r C 1 2 p/r m m 2 p/r s 1/p Y>s 1/p] ds 2 mp/r C 2 2 p/r 2 mp/r m/r Y >2 m/r] C 2 2 mp/r m/r Y >2 m/r] m 2 p/r C 3 2 m m/r E YI Y>2 m/r], if p<r, C 4 m2 m m/r E YI Y>2 m/r], if p r, C 5 2 mp/r m/r E YI Y>2 m/r], if p>r For the case p<r,if0<r<1, then 2 m m/r E YI Y>2 m/r] 2 m m/r m 2 /r <Y 2 /r] 2 /r <Y 2 /r] 2 1/r C 1 2 /r <Y 2 /r] 3.29 C 1 EY.

17 Discrete Dynamics in Nature and Society 17 If r 1, then 2 m m/r E YI Y>2 m/r] EYIY >2 m m 2 <Y 2 ] 2 <Y 2 ] 1 2 <Y 2 ] C 1 E Y log1 YI 2 <Y 2 ] 3.30 C 1 E Y log1 Y ]. Otherwise for r>1, it has 2 m m/r E YI Y>2 m/r] 2 m m/r m 2 /r <Y 2 /r] 2 /r <Y 2 /r] 2 m m/r C 1 2 /r 2 /r <Y 2 /r] C 1 E Y r I 2 /r <Y 2 /r] C 1 EY r Similarly, for the case p r, if0<r<1, then m2 m m/r E YI Y>2 m/r] m2 m m/r m 2 /r <Y 2 /r] 2 /r <Y 2 /r] 2 /r <Y 2 /r] C 1 2 /r <Y 2 /r] C 2 E Y log1 Y ]. m2 1/r 2 1/r 3.32

18 18 Discrete Dynamics in Nature and Society If r 1, then m2 m m/r E YI Y>2 m/r] m m 2 <Y 2 ] 2 <Y 2 ] m C <Y 2 ] C 2 E Ylog 2 1 YI 2 <Y 2 ] ] C 2 E Ylog 2 1 Y Otherwise, for r>1, it follows m2 m m/r Y >2 m/r] m2 m m/r 2 /r <Y 2 /r] m 2 /r <Y 2 /r] 2 /r <Y 2 /r] m2 m m/r 2 m m/r C 1 2 /r 2 /r <Y 2 /r] C 2 E Y r log1 Y ] On the other hand, for the case p>r,if0<p<1, then 2 mp/r m/r E YI Y>2 m/r] 2 mp 1/r m 2 /r <Y 2 /r] 2 /r <Y 2 /r] C 1 2 /r <Y 2 /r] 2 mp 1/r 3.35 C 1 EY.

19 Discrete Dynamics in Nature and Society 19 If p 1, then 2 mp/r m/r E YI Y>2 m/r] m 2 /r <Y 2 /r] 2 /r <Y 2 /r] 1 2 /r <Y 2 /r] C 1 EY log1 Y For p>1, it has 2 mp/r m/r E YI Y>2 m/r] 2 mp 1/r m 2 /r <Y 2 /r] 2 /r <Y 2 /r] 2 mp 1/r C 1 2 p 1/r 2 /r <Y 2 /r] C 1 E Y p I 2 /r <Y 2 /r] C 1 EY p Consequently, by 3.28, the conditions of Theorem 2.3 and inequalities above, we obtain that C 1 2 m m/r E YI Y>2 m/r], if p<r, H 1 C 2 m2 m m/r E YI Y>2 m/r], if p r, C 3 2 mp/r m/r E YI Y>2 m/r], if p>r

20 20 Discrete Dynamics in Nature and Society C 4 EY <, if 0 <r<1, for p<r, C 5 E Y log1 Y ] <, if r 1, C 6 EY r <, if r>1, C 7 E Y log1 Y ] <, if 0 <r<1, ] for p r, C 8 E Ylog 2 1 Y <, if r 1, C 9 E Y r log1 Y ] <, if r>1, C 10 EY <, if 0 <p<1, for p>r, C 11 E Y log1 Y ] <, if p 1, C 12 EY p <, if p> Since {Ỹ sj, 1 j< } is a mean zero AANA sequence and EỸ 2 sj EY 2 sj, Y 2 sj Y 2 j I Y j s 1/p s 2/p I Y j >s 1/p, j 1, similar to the proof of 3.11, weobtainthat H 2 C 1 2 p/r s 2/p E 2 max p/r 1 n 2 in i1 ji1 2 Ỹ sj ds 2 C 1 2 p/r s 2/p supe 2 p/r i1 i 1 max in 2 Ỹ sj 1 n 2 ji1 ds i2 C 2 2 p/r s 2/p sup EỸ 2 sj ds 2 p/r i 1 ji1 C 3 2 p/r s 2/p E Y 2 I Y s 1/p] ds 2 p/r C 4 2 p/r P Y>s 1/p ds 2 p/r 3.39 : C 3 H 21 C 4 H 22. Similar to the proof of Theorem 1.1 of Chen and Gan 7, byp < 2 and the conditions of Theorem 2.3, we have that H 21 2 p/r m m 2 p/r s 2/p E Y 2 I Y s 1/p] ds 2 mp/r 2 p/r 2 mp/r 2m/r E Y 2 I Y 2 /r]

21 Discrete Dynamics in Nature and Society 21 2 mp 2/r E Y 2 I Y 2 /r] m 2 p/r C 1 2 mr 2/r E Y 2 I Y 2 /r], if p<r, C 2 m2 mr 2/r E Y 2 I Y 2 /r], if p r, C 3 2 mp 2/r E Y 2 I Y 2 /r], if p>r C 4 EY r <, if p<r, C 5 E Y r log1 Y ] <, if p r, C 6 EY p <, if p>r On the other hand, by the proof of 3.28 and 3.38, it follows H 22 2 p/r s 1/p Y>s 1/p] ds 2 p/r C 1 EY <, if 0 <r<1, for p<r, C 2 E Y log1 Y ] <, if r 1, C 3 EY r <, if r>1, C 4 E Y log1 Y ] <, if 0 <r<1, ] for p r, C 5 E Ylog 2 1 Y <, if r 1, C 6 E Y r log1 Y ] <, if r>1, C 7 EY <, if 0 <p<1, for p>r, C 8 E Y log1 Y ] <, if p 1, C 9 EY p <, if p> Similar to the proof of 3.23, by the property EY j 0 and the proofs of 3.28 and 3.38,one has H p/r s 1/p max 2 p/r 1 n 2 1 n 2 i1 in i1 ji1 EY sj ds in 8 2 p/r s 1/p max E ] Yj I Yj >s 1/p ds 2 p/r ji1

22 22 Discrete Dynamics in Nature and Society C 1 2 p/r s 1/p Y>s 1/p] ds 2 p/r C 1 EY <, if 0 <r<1, for p<r, C 2 E Y log1 Y ] <, if r 1, C 3 EY r <, if r>1, C 4 E Y log1 Y ] <, if 0 <r<1, ] for p r, C 5 E Ylog 2 1 Y <, if r 1, C 6 E Y r log1 Y ] <, if r>1, C 7 EY <, if 0 <p<1, for p>r, C 8 E Y log1 Y ] <, if p 1, C 9 EY p <, if p> Consequently, by 3.25, 3.27, 3.28, 3.38, 3.39, 3.40, 3.41, and3.42, we finally obtain 2.6. Remar 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 C 2 m m/r E Y 1 I Y 1 > 2 m/r], if p<r, I 1 C m2 m m/r E Y 1 I Y 1 > 2 m/r], if p r, C 2 mp/r m/r E Y 1 I Y 1 > 2 m/r], if p>r, CE Y 1 r <, if p<r, CE Y 1 r log1 Y 1 ] <, if p r, CE Y 1 p <, if p>r, 3.43 where 1 r<2andp>0. For the case p<r, by taing r 1, we cannot get that 2 m m/r E Y 1 I Y 1 > 2 m/r] E Y 1 I Y 1 > 2 m CE Y 1 <. 3.44

23 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 y 2 ln 2 y, y > 2, C 2 ] 1 1 y 2 ln 2 y dy Obviously, it can be found that EY C 2 1 dy <. y ln y But for r 1, 2 m m/r E YI Y>2 m/r] E YI nm 2 C n 2 n <Y 2 ] 1 dy C 2 n yln 2 y 2 n <Y 2 ] n 1 1. ln 2 n Acnowledgments 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 wor is supported by the NNSF of China , HSSPF of the Ministry of Education of China 10YJA910005, Mathematical Tianyuan Youth Fund , Provincial Natural Science Research Project of Anhui Colleges KJ2010A005, Natural Science Foundation of Anhui Province QA03, and Academic Innovation Team of Anhui University KJTD001B. References 1 I. A. Ibragimov, Some limit theorems for stationary processes, Theory of Probability and Its Applications, vol. 7, no. 4, pp , R. M. Burton and H. Dehling, Large deviations for some wealy dependent random processes, Statistics & Probability Letters, vol. 9, no. 5, pp , D. L. Li, M. B. Rao, and X. C. Wang, Complete convergence of moving average processes, Statistics & Probability Letters, vol. 14, no. 2, pp , L. X. Zhang, Complete convergence of moving average processes under dependence assumptions, Statistics and Probability Letters, vol. 30, no. 2, pp , 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 , 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 , P. Y. Chen and S. X. Gan, On moments of the maximum of normed partial sums of ρ-mixing random variables, Statistics & Probability Letters, vol. 78, no. 10, pp , 2008.

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