Stochastic dominance for linear combinations of infinite-mean risks

TL;DR

This paper develops a new framework for comparing linear combinations of iid heavy-tailed risks using stochastic dominance, introducing a novel distribution class and analyzing compound Poisson sums.

Contribution

It introduces a new distribution class and establishes stochastic dominance conditions for linear combinations of infinite-mean risks, extending to compound Poisson sums and stable distributions.

Findings

A sufficient condition for stochastic dominance of linear combinations of iid risks.

Introduction of a new distribution class encompassing many heavy-tailed distributions.

Demonstration that compound Poisson summands belong to this class under dominance relations.

Abstract

In this paper, we establish a sufficient condition to compare linear combinations of independent and identically distributed (iid) infinite-mean random variables under usual stochastic order. We introduce a new class of distributions that includes many commonly used heavy-tailed distributions and show that within this class, a linear combination of random variables is stochastically larger when its weight vector is smaller in the sense of majorization order. We proceed to study the case where each random variable is a compound Poisson sum and demonstrate that if the stochastic dominance relation holds, the summand of the compound Poisson sum belongs to our new class of distributions. Additional discussions are presented for stable distributions.