Generalized Marcinkiewicz Laws for Weighted Dependent Random Vectors in Hilbert Spaces

Abstract

Theory of Probability &Its Applications, Volume 67, Issue 3, Page 434-451, November 2022.
The aim of this paper is to apply the theory of regularly varying functions for studying Marcinkiewicz weak and strong laws of large numbers for the weighted sum $S_n=\sum_{j=1}^{m_n}c_{nj}X_j$, where $(X_n;\, n\geq 1)$ is a sequence of dependent random vectors in Hilbert spaces, and $(c_{nj})$ is an array of real numbers. Moreover, these results are applied to obtain some results on the convergence of multivariate Pareto--Zipf distributions and multivariate log-gamma distributions.