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Further Developments on Stochastic Dominance for Different Classes of Infinite-mean Distributions

Keyi Zeng, Zhenfeng Zou, Yuting Su, Taizhong Hu

TL;DR

This paper advances the theory of stochastic dominance for infinite-mean, heavy-tailed risks by systematically examining the inclusion and interaction of four distribution classes $\mathcal{H}$, $\mathcal{V}$, $\mathcal{H}^*$, and $\mathcal{G}$, which capture key tail and transform properties. It clarifies how these classes relate under scaling, power and convex transforms, and sums, and it establishes when diversification via weighted sums preserves first-order dominance over individual components. The work extends known results to more practical models, including losses triggered by rare events, truncated variables, and compound-binomial constructions, and provides necessary and sufficient conditions for preserving $(SD)$ and $(SD^*)$ in these settings. By presenting counterexamples and diagrams, it also delineates the limits of these inclusions and the non-closure under convolution, contributing to both theoretical understanding and applied risk assessment where infinite-mean tails are relevant. Overall, the results offer a rigorous framework for evaluating diversification and tail risk across a broad class of heavy-tailed distributions.

Abstract

In recent years, stochastic dominance for independent and identically distributed (iid) infinite-mean random variables has received considerable attention. The literature has identified several classes of distributions of nonnegative random variables that encompass many common heavy-tailed distributions. A key result demonstrates that the weighted sum of iid random variables from these classes is stochastically larger than any individual random variable in the sense of the first-order stochastic dominance. This paper systematically investigates the properties and inclusion relationships among these distribution classes, and extends some existing results to more practical scenarios. Furthermore, we analyze the case where each random variable follows a compound binomial distribution, establishing necessary and sufficient conditions for the preservation of the aforementioned stochastic dominance relation.

Further Developments on Stochastic Dominance for Different Classes of Infinite-mean Distributions

TL;DR

This paper advances the theory of stochastic dominance for infinite-mean, heavy-tailed risks by systematically examining the inclusion and interaction of four distribution classes , , , and , which capture key tail and transform properties. It clarifies how these classes relate under scaling, power and convex transforms, and sums, and it establishes when diversification via weighted sums preserves first-order dominance over individual components. The work extends known results to more practical models, including losses triggered by rare events, truncated variables, and compound-binomial constructions, and provides necessary and sufficient conditions for preserving and in these settings. By presenting counterexamples and diagrams, it also delineates the limits of these inclusions and the non-closure under convolution, contributing to both theoretical understanding and applied risk assessment where infinite-mean tails are relevant. Overall, the results offer a rigorous framework for evaluating diversification and tail risk across a broad class of heavy-tailed distributions.

Abstract

In recent years, stochastic dominance for independent and identically distributed (iid) infinite-mean random variables has received considerable attention. The literature has identified several classes of distributions of nonnegative random variables that encompass many common heavy-tailed distributions. A key result demonstrates that the weighted sum of iid random variables from these classes is stochastically larger than any individual random variable in the sense of the first-order stochastic dominance. This paper systematically investigates the properties and inclusion relationships among these distribution classes, and extends some existing results to more practical scenarios. Furthermore, we analyze the case where each random variable follows a compound binomial distribution, establishing necessary and sufficient conditions for the preservation of the aforementioned stochastic dominance relation.

Paper Structure

This paper contains 3 sections, 7 theorems, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Properties of distribution classes

Key Result

Theorem 1.1

CHSZ25 Let $\bm X=(X_1, \ldots, X_n)$ be a vector of iid random variables with a common distribution function $F$. If $F\in\mathcal{H}$, then for $\bm \theta, \bm \eta\in [0,\infty)^n$ such that $\bm\theta \preceq_{\rm m} \bm \eta$. If $F\in\mathcal{H}^\ast$, then ${\rm ({\color{red} SD^*})}$ holds for all $\bm\theta\in [0,\infty)^n$.

Figures (4)

  • Figure 1: The function $\eta_1(x)$
  • Figure 2: The function $\eta_2(x)$
  • Figure 3: Venn diagram illustrating the relationships among the classes $\mathcal{H}$, $\mathcal{G}$, $\mathcal{V}$, and $\mathcal{H}^*$. The largest class $\mathcal{H}^*$ is indicated by the dashed boundary, while $\mathcal{H}$ is a subset of $\mathcal{V}$, and $\mathcal{G}$ has non-empty intersections with both $\mathcal{H}$ and $\mathcal{V}$.
  • Figure 4: Venn diagram illustrating the relationships among four classes $\mathcal{S}_{\rm P}$, $\mathcal{S}_{\rm F}$, $\mathcal{S}_{\rm C}$ and $\mathcal{H}^\ast$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Remark 2.2: Essential infimum
  • Proposition 3.1
  • Example 3.2
  • ...and 16 more