On the complete convergence for sequences of dependent random variables via stochastic domination conditions and regularly varying functions theory

TL;DR

This paper extends Rio's proof to establish complete convergence for sums of dependent random variables with regularly varying normalizing constants, covering various dependence structures and improving existing results under optimal moment conditions.

Contribution

It introduces new complete convergence results for dependent sequences, including $m$-pairwise negatively dependent and $m$-extended negatively dependent variables, under optimal moment conditions.

Findings

First complete convergence results for $m$-pairwise negatively dependent variables.

First results for $m$-extended negatively dependent variables.

Unified and improved convergence results for $ ext{mixing}$ sequences.

Abstract

This note develops Rio's proof [C. R. Math. Acad. Sci. Paris, 1995] of the rate of convergence in the Marcinkiewicz--Zygmund strong law of large numbers to the case of sums of dependent random variables with regularly varying normalizing constants. It allows us to obtain a complete convergence result for dependent sequences under uniformly bounded moment conditions. This result is new even when the underlying random variables are independent. The main theorems are applied to three different dependence structures: (i) $m$-pairwise negatively dependent random variables, (ii) $m$-extended negatively dependent random variables, and (iii) $\varphi$-mixing sequences. To our best knowledge, the results for cases (i) and (ii) are the first results in the literature on complete convergence for sequences of $m$-pairwise negatively dependent random variables and $m$-extended negatively dependent random variables under the optimal moment conditions even when $m=1$. While the results for cases (i) and (iii) unify and improve many existing ones, the result for case (ii) complements the main result of Chen et al. [J. Appl. Probab., 2010]. Affirmative answers to open questions raised by Chen et al. [J. Math. Anal. Appl., 2014] and Wu and Rosalsky [Glas. Mat. Ser. III, 2015] are also given. An example illustrating the sharpness of the main result is presented.