Complete moment convergence of moving average processes for m-WOD sequence

In this paper, the complete moment convergence for the partial sum of moving average processes {Xn=∑i=−∞∞aiYi+n,n≥1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{X_{n}=\sum_{i=-\infty }^{\infty }a_{i}Y_{i+n},n\geq 1\}$\end{document} is established under some mild conditions, where {Yi,−∞<i<∞}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{Y_{i},-\infty < i<\infty \}$\end{document} is a sequence of m-widely orthant dependent (m-WOD, for short) random variables which is stochastically dominated by a random variable Y, and {ai,−∞<i<∞}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{a_{i},-\infty < i<\infty \}$\end{document} is an absolutely summable sequence of real numbers. These conclusions promote and improve the corresponding results from m-extended negatively dependent (m-END, for short) sequences to m-WOD sequences.


Introduction and main results
Let {Y i , -∞ < i < ∞} be a sequence of random variables and {a i , -∞ < i < ∞} be an absolutely summable sequence of real numbers, and for n ≥ 1 set X n = ∞ i=-∞ a i Y i+n . The limit properties of the moving average process {X n , n ≥ 1} have been extensively investigated by many authors. For example, Burton and Dehling [1] obtained a large deviation principle, Ibragimov [2] established the central limit theorem, Račkauskas and Suquet [3] proved the functional central limit theorems for self-normalized partial sums of linear processes, and An [4], Chen et al. [5], Kim and Ko [6], Li et al. [7], Li and Zhang [8], Wang and Hu [9], Yang and Hu [10], Zhang [11], Zhou [12], Zhou and Lin [13], Zhang [14], Zhang and Ding [15], Song and Zhu [16,17] got the complete (moment) convergence of moving average process based on a sequence of different dependent (or mixing) random variables, respectively. But few results for moving average process based on m-WOD random variables are known. Firstly, we introduce some definitions. Definition 1.1 A sequence {Y i , -∞ < i < ∞} of random variables is said to be stochastically dominated by a random variable Y if there exists a constant C such that The concept of widely orthant dependence structure was introduced by Wang et al. [18] as follows. Definition 1. 3 For the random variables {X n , n ≥ 1}, if there exists a finite positive sequence {g U (n), n ≥ 1} satisfying, for each n ≥ 1 and for all x i ∈ R, 1 ≤ i ≤ n, then we say that the random variables {X n , n ≥ 1} are widely lower orthant dependent (WLOD, for short); if they are both WUOD and WLOD, then we say that the random variables {X n , n ≥ 1} are widely orthant dependent (WOD, for short), and g U (n), g L (n), n ≥ 1, are called dominated coefficients.
Inspired by WOD and m-NA, Fang et al. [19] introduced the following notion.

Definition 1.4
Let m ≥ 1 be a fixed integer. A sequence of random variables {X n , n ≥ 1} is said to be m-WOD if, for any n ≥ 2 and i 1 , i 2 , . . . , i n such that |i ki j | ≥ m for all 1 ≤ k = j ≤ n, we have that X i 1 , X i 2 , . . . , X i n are WOD.
By (1.1) and (1.2), we can see that g U (n) ≥ 1 and g L (n) ≥ 1. Recall that when g U (n) = g L (n) = M for some positive constant M and any n ≥ 1, then the random variables {X n , n ≥ 1} are called extended negatively dependent (END, for short). The definition of END was introduced by Liu [20]. If both (1.1) and (1.2) hold for g U (n) = g L (n) = 1 for any n ≥ 1, then the random variables {X n , n ≥ 1} are called negatively orthant dependent (NOD, for short), which was introduced by Ebrahimi and Ghosh [21]. It is well known that negatively associated (NA, for short) random variables are NOD. Hu [22] pointed out that negatively superadditive dependent (NSD, for short) random variables are NOD. Hence, the class of m-WOD random variables includes independent sequence, m-NA sequence, NSD sequence, m-NOD sequence, and m-END sequence as special cases. Studying the probability limit theory and its applications for m-WOD random variables is of great interest. But there are few results on the complete moment convergence of moving average process based on an m-WOD sequence. Therefore, in this paper, we establish some results on the complete moment convergence for partial sums for moving average process.
Throughout the sequel, C represents a positive constant although its value may change from one appearance to the next, I{A} denotes the indicator function of the set A, [x] denotes the integer part of x, X + = max{X, 0}, X -= max{-X, 0}.

Preliminary lemmas
In this section, we give some lemmas which will be useful to prove our main results. Lemma 2.1 (Fang et al. [19]) Let {X n , n ≥ 1} be a sequence of m-WOD random variables with dominating coefficients g(n) = max{g L (n), g U (n)}). If {f n (·), n ≥ 1} are all nondecreasing (or nonincreasing), then {f n (X n ), n ≥ 1} are still m-WOD with dominating coefficients {g(n), n ≥ 1}. Lemma 2.2 (Fang et al. [19]) For a positive real number q ≥ 2, if {X n , n ≥ 1} is a sequence of mean zero m-WOD random variables with dominating coefficients g(n) = max{g L (n), g U (n)}. If E|X i | q < ∞ for every i ≥ 1, then for all n ≥ 1 there exist positive constants C 1 (m, q) and C 2 (m, q) depending on q and m such that
Lemma 2.4 (Wang et al. [23]) Let {X n , n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X. Then, for any a > 0 and b > 0,
For the complete convergence, we have the following corollary from the above theorems immediately. Under the assumptions of Theorem 3.2, for any ε > 0, we have ∞ n=1 n -1 l(n)P n j=1 X j > εn 1/p < ∞. (3.14) Remark 3.4 Since m-WOD random variables include independent, m-NA, NSD, WOD, m-NOD, and m-END random variables, so our results also hold for independent, m-NA, NSD, WOD, m-NOD, and m-END random variables, and therefore Theorem 3.1 and Theorem 3.2 improve upon the known results.
Remark 3.5 Obviously, the assumption that {Y i , -∞ < i < ∞} is stochastically dominated by a random variable Y is weaker than the assumption of identical distribution of the random variables {Y i , -∞ < i < ∞}, therefore the results of Theorem 3.1 and Theorem 3.2 also hold for identically distributed random variables.
Remark 3.6 Let a 0 = 1, a i = 0, i = 0, then S n = n k=1 X k = n k=1 Y k . Hence the results of Theorem 3.1 and Theorem 3.2 also hold when {X k , k ≥ 1} is a sequence of m-WOD random variables which is stochastically dominated by a random variable Y .
Remark 3.7 The results obtained by this paper and Fang et al. [19] are different. In our paper, we mainly discuss the complete moment convergence of moving average processes for an m-WOD sequence, Fang et al. [19] proved the asymptotic approximations of ratio moments based on the m-WOD sequence.

Conclusions
In this paper, using the moment inequality for m-WOD sequences and truncation method, the complete moment convergence for the partial sum of moving average processes {X n = ∞ i=-∞ a i Y i+n , n ≥ 1} is established, where {Y i , -∞ < i < ∞} is a sequence of m-WOD random variables which is stochastically dominated by a random variable Y , and {a i , -∞ < i < ∞} is an absolutely summable sequence of real numbers. These conclusions obtained extend and improve the corresponding results from m-END sequences to m-WOD sequences.