On strong law of large numbers for non identically distributed random variables

TL;DR

This paper introduces a new strong law of large numbers applicable to sequences of random variables that are mostly independent with some dependencies, relaxing the need for finite expectations by allowing moments to approach zero.

Contribution

It proposes a version of the strong law of large numbers that accommodates dependent variables and relaxes the expectation requirement, broadening applicability.

Findings

Validates the law for sequences with small dependent parts

Allows moments of summands to tend to zero

Extends classical results to more general cases

Abstract

A new version of a strong law of large numbers for a ``good'' pairwise independent sequence of random variables (r.v.'s) with a small part of ``bad'' dependent r.v.'s is proposed. The main goal is to relax the assumption on the existence of the expectation for each summand: the members of an ``infrequent'' part of the whole sequence may have moments of orders converging to zero.