Complete convergence for the maximal partial sums without maximal inequalities

Abstract

This work provides the necessary and sufficient conditions for complete convergence for the maximal partial sums of dependent random variables. The results are proved without using maximal inequalities. The main theorems can be applied to sequences of (i) m-pairwise negatively dependent random variables and (ii) m-extended negatively dependent random variables. While the result for case (i) unifies and improves many existing ones, the result for case (ii) complements the main theorem 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. Two examples illustrating the sharpness of the main result are presented.