Research Article

Complete convergence for the maximal partial sums without maximal inequalities

Published in: Quaestiones Mathematicae
Volume 47 , issue 7 , 2024 , pages: 1387–1402

DOI: 10.2989/16073606.2024.2323150

Author(s): Fakhreddine Boukhari Abou Bekr Belkaid University, Algeria , Nguyen Chi Dzung Institute of Mathematics, Vietnam Academy of Science and Technology, Vietnam , Lê Vǎn Thaǹh Vinh University, Vietnam

Keywords: 60F15 , Complete convergence , rate of convergence , maximal inequality , dependent random variables , regularly varying function

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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.

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