Further Developments on Stochastic Dominance for Convex Combinations of Infinite-Mean Random Variables

TL;DR

This paper explores stochastic dominance properties for sums of iid infinite-mean random variables, extending results to compound binomial cases and clarifying distribution class relationships.

Contribution

It systematically analyzes distribution class relationships and extends stochastic dominance results to compound binomial distributions for practical applications.

Findings

Weighted sums of iid heavy-tailed variables are stochastically larger than individual variables.

The paper establishes conditions for stochastic dominance preservation in compound binomial cases.

It clarifies inclusion relationships among distribution classes for infinite-mean variables.

Abstract

In recent years, stochastic dominance for independent and identically distributed (iid) infinite-mean random variables has received considerable attention. The literature has identified several classes of distributions of nonnegative random variables that encompass many common heavy-tailed distributions. A key result demonstrates that the weighted sum of iid random variables from these classes is stochastically larger than any individual random variable in the sense of the first-order stochastic dominance. This paper systematically investigates the properties and inclusion relationships among these distribution classes, and extends some existing results to more practical scenarios. Furthermore, we analyze the case where each random variable follows a compound binomial distribution, establishing necessary and sufficient conditions for the preservation of the aforementioned stochastic dominance relation.