Summability of Sequence of Random Variables

TL;DR

This paper investigates the summability properties of double sequences of random variables, establishing regularity results for various modes of convergence and extending the theory to include extended real-valued variables.

Contribution

It introduces new results on the regularity of summability methods for sequences of random variables, including those with extended real values, and develops a novel multiplication construction for infinite matrices.

Findings

Regular summability methods preserve almost everywhere, almost sure, and Lp-convergence.

Summability methods are not necessarily regular for convergence in probability.

Extension of summability theory to extended real-valued random variables.

Abstract

In this paper, we study the summability properties of double sequences of real constants which map sequences of random variables to sequences of random variables that are defined on the same probability sample space. We show that a regular method of summability is still regular on sequences of random variables with almost everywhere convergence, almost sure convergence, and with $L_p$-convergence. It is not necessarily regular on sequences of random variables with convergence in probability. We extend these results to random variables with values in extended real numbers (extended real numbers include infinite values, see definitions 2.2 and 2.3). For this we introduce a construction that allows us to multiply sequences of extended real numbers with infinite real matrices.