Risk aggregation and stochastic dominance for a class of heavy-tailed distributions

TL;DR

This paper introduces a new class of heavy-tailed distributions where weighted averages of i.i.d. variables dominate individual variables in stochastic order, with implications for risk aggregation involving distributions like Pareto and Fréchet.

Contribution

The paper characterizes a new class of heavy-tailed distributions with stochastic dominance properties for weighted averages, extending to dependent and non-identically distributed variables.

Findings

Many heavy-tailed distributions belong to this class.

Weighted averages dominate individual variables in stochastic order.

Results extend to dependent and non-identically distributed variables.

Abstract

We introduce a new class of heavy-tailed distributions for which any weighted average of independent and identically distributed random variables is larger than one such random variable in (usual) stochastic order. We show that many commonly used extremely heavy-tailed (i.e., infinite-mean) distributions, such as the Pareto, Fr\'echet, and Burr distributions, belong to this class. The established stochastic dominance relation can be further generalized to allow negatively dependent or non-identically distributed random variables. In particular, the weighted average of non-identically distributed random variables dominates their distribution mixtures in stochastic order.