Randomly stopped sums with consistently varying distributions

TL;DR

This paper investigates conditions under which the distribution of randomly stopped sums with independent, not necessarily identically distributed summands, belongs to the class of consistently varying distributions, expanding understanding of their probabilistic behavior.

Contribution

It provides new conditions for the distribution of randomly stopped sums with varying distributions to be consistently varying, without requiring identical distribution assumptions.

Findings

Identifies conditions for consistently varying distributions in random sums.

Extends previous results to non-identically distributed summands.

Provides theoretical framework for analyzing such sums.

Abstract

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables, and $\eta$ be a counting random variable independent of this sequence. We consider conditions for $\{\xi_1,\xi_2,\ldots\}$ and $\eta$ under which the distribution function of the random sum $S_{\eta}=\xi_1+\xi_2+\cdots+\xi_{\eta}$ belongs to the class of consistently varying distributions. In our consideration, the random variables $\{\xi_1,\xi_2,\ldots\}$ are not necessarily identically distributed.