Intermediately Trimmed Sums of Oppenheim Expansions: a Strong Law

TL;DR

This paper establishes strong laws for intermediately trimmed sums of random variables with infinite means, including cases with dependent variables via Oppenheim expansions, extending previous results in the field.

Contribution

It introduces new convergence results for intermediately trimmed sums, generalizing prior work and applying to dependent variables through Oppenheim expansions.

Findings

Almost sure convergence of trimmed sums under new conditions

Extension to dependent variables via Oppenheim expansions

First study of generalized Oppenheim expansions in this context

Abstract

The work of this paper is devoted to obtaining strong laws for intermediately trimmed sums of random variables with infinite means. Particularly, we provide conditions under which the intermediately trimmed sums of independent but not identically distributed random variables converge almost surely. Moreover, by dropping the assumption of independence we provide a corresponding convergence result for a special class of Oppenheim expansions. We highlight that the results of this paper generalize the results provided in the recent work of \cite{KS} while the convergence of intermediately trimmed sums of generalized Oppenheim expansions is studied for the first time.