Paper: arXiv 2505.01739

Authors: Yuyu Chen, Taizhong Hu, Seva Shneer, Zhenfeng Zou

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous mathematical proofs establishing a new class of distributions (H) for stochastic dominance under majorization for infinite-mean risks. However, as a purely theoretical probability/actuarial science manuscript, it lacks empirical validation, backtesting, or historical financial data applications.

Research Flowchart

  flowchart TD
    A[Research Goal: Stochastic Dominance in Linear Combinations of Infinite-Mean iid Risks] --> B[Input: Majorization Order of Weight Vectors & Heavy-Tailed Distributions]
    
    subgraph Methodology & Processing
        B --> C[Define New Class of Heavy-Tailed Infinite-Mean Distributions]
        C --> D[Analyze Usual Stochastic Order under Majorized Weights]
        D --> E[Extend Framework to Compound Poisson Sums & Stable Distributions]
    end

    E --> F[Key Outcome 1: Smaller Weights in Majorization Order Yield Stochastically Larger Outcomes]
    E --> G[Key Outcome 2: Necessary & Sufficient Conditions for Summands in Compound Poisson Sums]