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

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.