On Complete Convergence in Mean for Double Sums of Independent Random Elements in Banach Spaces

TL;DR

This paper establishes conditions for complete convergence in mean of double sums of independent random elements in Banach spaces, characterizing Rademacher type p spaces and linking to strong laws of large numbers.

Contribution

It provides necessary and sufficient conditions for complete convergence in mean in Banach spaces, characterizing Rademacher type p spaces and connecting to classical probability results.

Findings

Characterization of Rademacher type p Banach spaces via convergence conditions

Conditions under which double sums converge completely in mean of order p

Link between convergence in mean and strong law of large numbers in non-Rademacher type spaces

Abstract

In this work, conditions are provided under which a normed double sum of independent random elements in a real separable Rademacher type $p$ Banach space converges completely to $0$ in mean of order $p$. These conditions for the complete convergence in mean of order $p$ are shown to provide an exact characterization of Rademacher type $p$ Banach spaces. In case the Banach space is not of Rademacher type $p$, it is proved that the complete convergence in mean of order $p$ of a normed double sum implies a strong law of large numbers.