Stochastic dominance for linear combinations of infinite-mean risks

Abstract

We establish a sufficient condition for comparing linear combinations of independent and identically distributed infinite-mean random variables under the usual stochastic order. We introduce a new class of distributions that includes many commonly used heavy-tailed models 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.