On weak laws of large numbers for maximal partial sums of pairwise independent random variables

TL;DR

This paper extends Rio's method to establish weak laws of large numbers for maximal partial sums of pairwise independent variables without relying on Kolmogorov's inequality, under a uniform integrability condition.

Contribution

It introduces a new proof technique for weak laws of large numbers applicable to pairwise independent variables, avoiding traditional inequalities.

Findings

Weak law established under uniform integrability.

Method avoids Kolmogorov maximal inequality.

Result's sharpness demonstrated with an example.

Abstract

This paper develops Rio's method [C. R. Acad. Sci. Paris S\'{e}r. I Math., 1995] to prove the weak law of large numbers for maximal partial sums of pairwise independent random variables. The method allows us to avoid using the Kolmogorov maximal inequality. As an application, a weak law of large numbers for maximal partial sums of pairwise independent random variables under a uniform integrability condition is also established. The sharpness of the result is illustrated by an example.