Stochastic dominance for linear combinations of infinite-mean risks

2025-05-03

AI Summary

This paper investigates stochastic order comparisons of linear combinations of i.i.d. heavy-tailed random variables with infinite means. Addressing distributions such as Pareto and stable laws, we establish the first sufficient conditions for usual stochastic dominance among such infinite-mean linear combinations. We introduce a novel parametric family of heavy-tailed distributions—encompassing many common ones—and prove that, when summands belong to this family, a smaller weight vector in the majorization order yields a stochastically larger linear combination under the usual stochastic order. Furthermore, we uncover an exact equivalence between this dominance property and the distributional structure of the summands in a compound Poisson framework, thereby characterizing precisely the class of distributions satisfying the dominance condition. Our results provide a rigorous theoretical foundation for ordering and managing aggregate risk in heavy-tailed settings.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Compare linear combinations of infinite-mean iid variables
• Introduce new class of heavy-tailed distributions for dominance analysis
• Study stochastic dominance in compound Poisson sums

Innovation

• Introduces new class of heavy-tailed distributions
• Compares linear combinations via majorization order
• Links compound Poisson summands to new class