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

Abstract

In this paper, the complete moment convergence for the partial sum of moving average processes $\{X_{n}=\sum_{i=-\infty }^{\infty }a_{i}Y_{i+n},n\geq 1\}$ { X n = ∑ i = − ∞ ∞ a i Y i + n , n ≥ 1 } is established under some mild conditions, where $\{Y_{i},-\infty < i<\infty \}$ { Y i , − ∞ < i < ∞ } is a sequence of m-widely orthant dependent (m-WOD, for short) random variables which is stochastically dominated by a random variable Y, and $\{a_{i},-\infty < i<\infty \}$ { a i , − ∞ < i < ∞ } 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.

Complete qth-Moment Convergence of Moving Average Process for m-WOD Random Variable

“…We get the results without slowly varying function, so our results in the paper extend and improve the results in Ref. [14].…”

“…Proof If r > 11 ≤ p < 2obviously r = rp(1/p), and 1/p > 1/2rp > 1so Lemma 4 satisfies the conditions of the Theorem 3.1 in Ref. [14].…”

“…There are few results on the complete qth-moment convergence of moving average process for m-WOD random variables. Therefore, in this paper, based on Guan 􀆳 s research [14] , we study the complete qth-moment convergence of moving average processes based on m-WOD random variables, the results extend and improve the corresponding ones under WOD, m-NA, m-NOD, m-END random variables.…”

“…Lemma 4 [14] Let r > 11 ≤ p < 2{X n n ≥ 1} be a moving average process under the sequence {Y n  -¥ < n < ¥} of m-WOD random variables with dominating coefficients g(n), {Y n  -¥ < n < ¥}  Y and EY j = 0 for every j, the real number sequence {a…”

“…Recently, some results have been obtained under the assumption that the sequence {Y n  -¥ < n < ¥} is dependent. For example, Li [9] investigated the convergence properties under ρ-mixing assumptions; Zhang [10] and Chen et al [11] established complete convergence under φ-mixing assumptions; Song and Zhu [12] got the complete convergence of moving average process based on ρ --mixing assumptions; Tao [13] discussed the complete convergence under WOD random variables; Guan [14] obtained the complete moment convergence under m-WOD random variables.…”