# Bulletin of theKorean Mathematical SocietyBKMS

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

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Bull. Korean Math. Soc.

Published online March 16, 2022

## ON COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF COORDINATEWISE NEGATIVELY ASSOCIATED RANDOM VECTORS IN HILBERT SPACES

Vu Thi Ngoc Anh and 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