Convergence of series of conditional expectations

TL;DR

This paper investigates convergence rates in the strong law of large numbers for Banach space valued random variables, applying martingale approximation to derive results for conditional expectations and Markov chains.

Contribution

It introduces new convergence rate results for partial sums of conditional expectations and powers of Markov chain operators using martingale methods.

Findings

Established convergence rates for Banach space valued sums

Applied results to weighted partial sums of conditional expectations

Extended analysis to powers of reversible Markov chain operators

Abstract

This paper deals with rates of convergence in the strong law of large numbers, in the Baum-Katz form, for partial sums of Banach space valued random variables. The results are then applied to solve similar problems for weighted partial sums of conditional expectations. They are further used to treat partial sums of powers of a reversible Markov chain operator. The method of proof is based on martingale approximation. The conditions are expressed in terms moments of the individual summands.