Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2022; 59(4): 879-895

Published online July 31, 2022 https://doi.org/10.4134/BKMS.b210509

Copyright © The Korean Mathematical Society.

On complete convergence for weighted sums of coordinatewise negatively associated random vectors in Hilbert spaces

Vu Thi Ngoc Anh, Nguyen Thi Thanh Hien

Hoa Lu University; Hanoi University of Science and Technology

Abstract

This paper establishes the {Baum--Katz} type theorem and the {Marcinkiewicz--Zymund} type strong law of large numbers for sequences of coordinatewise negatively associated and identically distributed random vectors $\{X,X_n,n\ge1\}$ taking values in a Hilbert space $H$ with general normalizing constants $b_n=n^{\alpha}\widetilde L(n^{\alpha})$, where $\widetilde L(\cdot)$ is the de Bruijn conjugate of a slowly varying function $L(\cdot).$ The main result extends and unifies many results in the literature. The sharpness of the result is illustrated by two examples.

Keywords: Weighted sum, negative association, Hilbert space, complete convergence, strong law of large numbers, slowly varying function

MSC numbers: 60F15