Strong laws of large numbers for intermediately trimmed sums of i.i.d. random variables with infinite mean

TL;DR

This paper establishes strong laws of large numbers for intermediately trimmed sums of i.i.d. non-negative random variables, including cases with infinite mean, by identifying appropriate trimming strategies.

Contribution

It introduces a method to determine proper moderate trimming that ensures strong laws of large numbers for distributions with infinite mean.

Findings

Existence of proper moderate trimming for all distributions

Necessary and sufficient conditions for regularly varying tails

Strong law of large numbers under specific trimming strategies

Abstract

We consider moderately trimmed sums of non-negative i.i.d. random variables. We show that for every distribution function there exists a proper moderate trimming such that for the trimmed sum a non-trivial strong law of large numbers holds. In case that the distribution function has regularly varying tails we give necessary and sufficient conditions on the trimming for a strong law of large numbers to hold.