Further research on complete moment convergence for moving average process of a class of random variables

In this article, the complete moment convergence for the partial sum of moving average processes \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}{Xn=∑i=−∞∞aiYi+n,n≥1} is established under some mild conditions, where \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}{Yi,−∞<i<∞} is a doubly infinite sequence of random variables satisfying the Rosenthal type maximal inequality and \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}{ai,−∞<i<∞} is an absolutely summable sequence of real numbers. These conclusions promote and improve the corresponding results given by Ko (J. Inequal. Appl. 2015:225, 2015).


Introduction
We first introduce the definition of the Rosenthal type maximal inequality, which is one of the most interesting inequalities in probability theory and mathematical statistics. Suppose that {Y n , n ≥ } is a sequence of random variables satisfying E|Y i | r < ∞ for r ≥ , then there exists a positive constant C(r) depending only on r such that The following definitions will be useful in this paper. The first one can be found in Kuczmaszewska [].
Definition . A sequence {Y i , -∞ < i < ∞} of random variables is said to satisfy a weak dominating condition with a dominating random variable Y if where C is a positive constant.
Definition . A real valued function l(x), positive and measurable on [, ∞), is said to be slowly varying at infinity if for each λ > , lim x→∞ Throughout the paper, let {Y i , -∞ < i < ∞} be a sequence of random variables with zero means and {a i , -∞ < i < ∞} be a sequence of real numbers with ∞ i=-∞ |a i | < ∞, and the moving average process {X n , n ≥ } is defined by X n = ∞ i=-∞ a i Y i+n . The complete moment convergence of moving average process {X n , n ≥ } has been widely investigated by many authors. We list some results as follows.
Li and Zhang [] established the following complete moment convergence of moving average processes under NA assumptions.
Later on, the following complete moment convergence of moving average processes generated by ρ-mixing sequence was proved by Zhou and Lin [].
Theorem B Let h be a function slowly varying at infinity, p ≥ , pα > , and α > /. Suppose that {X n , n ≥ } is a moving average process based on a sequence Recently, Ko [] obtained the complete moment convergence of moving average processes generated by a class of random variable.
Theorem C Let h be a function slowly varying at infinity, p ≥ , pα > , and α > /. Assume that {a i , -∞ < i < ∞} is an absolutely summable sequence of real numbers and that {Y i , -∞ < i < ∞} is a sequence of mean zero random variables satisfying a weak mean dominating condition with a mean dominating random variable Y and E|Y | p h(|Y | /α ) < ∞.
Suppose that {X n , n ≥ } is a moving average process based on the sequence {Y i , -∞ < i < ∞}. Assume that the Rosenthal type maximal inequality of Y xj = -xI{Y j < -x} + Y j I{|Y j | ≤ x} + xI{Y j > x} holds for any q ≥  and x > . Then, for all ε > , The aim of this paper is to study the complete moment convergence of moving average process of random sequence under the assumption that the random variables satisfy the Rosenthal type maximal inequality and the weak mean dominating condition. The paper is organized as follows. In Section  we describe the main results, Sections  and  provide some lemmas and the details of the proofs, respectively. Throughout the sequel, C represents a positive constant although its value may change from one place to the next, a n = O(b n ) means |a n /b n | ≤ C and I{A} stands for the indicator function of the set A.

Main results
Theorem . Let l be a function slowly varying at infinity. Suppose that {a i , -∞ < i < ∞} is an absolutely summable sequence of real numbers. Let {g(n), n ≥ } and {f (n), n ≥ } be two sequences of positive constants such that, for some r ≥ max{, p}, p ≥ ,

moving average process generated by a sequence of random variables {Y i , -∞ < i < ∞} with mean zeros and satisfying a weak dominating condition with a dominating random variable Y and E|Y
Assume that the Rosenthal type maximal inequality of Y xj = -xI{Y j < -x} + Y j I{|Y j | ≤ x} + xI{Y j > x} holds for the above r and all x > . Then, for all ε > , Corollary . If we replace conditions (C)-(C) by the following: . The other assumptions of Theorem . also hold, then, for all ε > , we have Conditions (C)-(C) can be satisfied by many sequences, we list some as the following remarks.
Remark . Let g(n) = n pα- , f (n) = n α for pα > , and / < α ≤ , assume that (.) holds true for {Y xj } and then conditions (C)-(C) can be verified easily by Lemma ., therefore we know Obviously, Theorem . and Corollary . from Ko [] are the same as (.) and (.), respectively, so we extend the known results. If we take a  = , for any j ≥ . Assume that the Rosenthal type maximal inequality of Y nj = Y j I{|Y j | ≤ f (n)} holds true for r = . Then, for all ε > ,

Preliminary lemmas
In order to prove the main results, we shall need the following lemmas.

Lemma . (Zhou []) If l is slowly varying at infinity, then ()
Then for any a >  and some constant C

Proofs
Proof of Theorem . Obviously that ) < ∞, then by Lemma . and condition (C), for any x > f (n), we conclude Therefore, one can get for any ε >  and x > f (n) large enough. Hence it follows that Now we want to estimate I  < ∞. It is obvious that |Y j -Y xj | ≤ |Y j |I{|Y j | > x}, then it follows by Markov's inequality, Lemma . and conditions (C) and (C) that Hence it remains to show that I  < ∞. By Markov's inequality, the Hölder inequality and the Rosenthal type maximal inequality, for r > max{, p}, it is easy to see that For I  , it follows by C r inequality, Lemma . and conditions (C), (C), and (C) that Finally we want to show that I  < ∞, by C r inequality, Lemma . and conditions (C), (C), and (C), it follows that Hence the proof of (.) is completed by combining (.)-(.).
Hence for n large enough and any ε > , we obtain By Markov's inequality, (.), and (.), it is easy to check that It follows from Markov's inequality, the Hölder inequality, the Rosenthal type inequality, (.), and (.) that Thus we have completed the proof of Theorem ..