A Refinement of the Kolmogorov-Marcinkiewicz-Zygmund Strong Law of Large Numbers

TL;DR

This paper refines the Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers by extending recent results on complete moment convergence to new cases and establishing almost sure convergence analogues, including in Banach spaces.

Contribution

It extends the theory of complete moment convergence and almost sure convergence for i.i.d. sequences with finite p-th moments, covering cases p=1 and 0<p<1, and generalizes to Banach spaces.

Findings

Extended complete moment convergence results to p=1 and 0<p<1.

Established almost sure convergence analogues for these cases.

Presented results applicable in Banach space settings.

Abstract

For the partial sums formed from a sequence of i.i.d. random variables having a finite absolute p'th moment for some p in (0,2), we extend the recent and striking discovery of Hechner and Heinkel (Journal of Theoretical Probability (2010)) concerning "complete moment convergence" to the two cases 0<p<1 and p=1. Moreover, for 0<p<2, we obtain "almost sure convergence" analogues of these "complete moment convergence" results and these "almost sure convergence" analogues may be regarded as being a refinement of the celebrated Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers. Versions of the above results in a Banach space setting are also presented.