On asymptotic scales of independently stopped random sums

TL;DR

This paper investigates the asymptotic behavior of independently stopped random sums, focusing on heavy-tailed distributions, and provides criteria to determine whether the sum's asymptotics are dominated by the process or the stopping variable.

Contribution

It introduces a unified framework using asymptotic scales for heavy-tailed variables to analyze stopped sums and derives conditions for moment determinacy.

Findings

Identifies when the sum's asymptotics are dominated by the increment or stopping variable.

Provides new sufficient conditions for moment determinacy of compounded sums.

Extends analysis to all heavy-tailed distributions using asymptotic scales.

Abstract

We study randomly stopped sums via their asymptotic scales. First, finiteness of moments is considered. To generalise this study, asymptotic scales applicable to the class of all heavy-tailed random variables are used. The stopping is assumed to be independent of the underlying process, which is a random walk. The main result enables one to identify whether the asymptotic behaviour of a stopped sum is dominated by the increment, or the stopping variable. As a consequence of this result, new sufficient conditions for the moment determinacy of compounded sums are obtained.