Stochastic dominance for linear combinations of infinite-mean risks
Yuyu Chen, Taizhong Hu, Seva Shneer, Zhenfeng Zou
TL;DR
This work tackles the problem of comparing linear combinations of iid random variables with infinite mean under the usual stochastic order by introducing a broad distribution class $\mathcal{H}$. The authors prove a main result: if $F\in\mathcal{H}$, then for any majorization $\overline{\theta}\preceq\overline{\eta}$ the sum $\sum_i \eta_i X_i$ is stochastically dominated by $\sum_i \theta_i X_i$, extending classical finite-mean results to heavy-tailed settings. They further show that for compound Poisson sums, the stochastic dominance holds if and only if the summand distribution lies in $\mathcal{H}$ (with extensions to $\mathcal{H}^*$ for stronger relations), and they discuss analogous implications for stable distributions. The paper thus broadens the toolkit for risk diversification and aggregation in infinite-mean contexts, offering new closure properties and a spectrum of heavy-tailed distributions that satisfy the desired order. It also outlines several open questions, including necessity without lower-boundedness and dependence structures beyond independence.
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.