Complete Moment Convergence of Moving Average Processes Generated by Negatively Associated Sequences

Title & Authors
Complete Moment Convergence of Moving Average Processes Generated by Negatively Associated Sequences
Ko, Mi-Hwa;

Abstract
Let {$\small{X_i,-{\infty}}$ < 1 < $\small{\infty}$} be a doubly infinite sequence of identically distributed and negatively associated random variables with mean zero and finite variance and {$\small{a_i,\;-{\infty}}$ < i < $\small{{\infty}}$} be an absolutely summable sequence of real numbers. Define a moving average process as $\small{Y_n={\sum}_{i=-\infty}^{\infty}a_{i+n}X_i}$, n $\small{\geq}$ 1 and $\small{S_n=Y_1+{\cdots}+Y_n}$. In this paper we prove that E|$\small{X_1}$|$\small{^rh}$($\small{|X_1|^p}$) < $\small{\infty}$ implies $\small{{\sum}_{n=1}^{\infty}n^{r/p-2-q/p}h(n)E{max_{1{\leq}k{\leq}n}|S_k|-{\epsilon}n^{1/p}}{_+^q}}$<$\small{{\infty}}$ and $\small{{\sum}_{n=1}^{\infty}n^{r/p-2}h(n)E{sup_{k{\leq}n}|k^{-1/p}S_k|-{\epsilon}}{_+^q}}$<$\small{{\infty}}$ for all $\small{{\epsilon}}$ > 0 and all q > 0, where h(x) > 0 (x > 0) is a slowly varying function, 1 $\small{{\leq}}$ p < 2 and r > 1 + p/2.
Keywords
Moving average process;negatively associated;complete moment convergence;doubly infinite sequence;
Language
English
Cited by
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