ON THE COMPLETE MOMENT CONVERGENCE OF MOVING AVERAGE PROCESSES GENERATED BY ρ^*-MIXING SEQUENCES

Abstract

Let {Y i ; −∞ < i < ∞} be a doubly infinite sequence of identically distributed and ρ *-mixing random variables with zero means and finite variances and {a i ; −∞ < i < ∞} an absolutely summable sequence of real numbers. In this paper, we prove the complete moment convergence of { P n k=1 P ∞ i=−∞ a i+k Y i /n 1/p ; n ≥ 1} under some suitable conditions. We extend Theorem 1.1 of Li and Zhang [Y. X. Li and L. X. Zhang, Complete moment convergence of moving average processes under dependence assumptions, Statist. Probab. Lett. 70 (2004), 191-197.] to the ρ *-mixing case.

Convergence properties of the maximum partial sums for moving average process under ρ − $\rho^{-}$ -mixing assumption

“…Recently, several results have been obtained under the assumption that the sequence {Y n , -∞ < n < +∞} is dependent. For example, Zhang [7] and Chen et al [8] discussed the limiting behavior of moving average processes under ϕ-mixing assumptions; Baek et al [9] and Yu and Wang [10] established complete convergence under NA assumptions; Li and Zhang [11], Kim and Ko [12][13][14] and Zhou [15,16] discussed the complete moment convergence under NA, ϕmixing and ρ-mixing random variables, and Qiu and Chen [17] obtained the complete moment convergence under END.…”