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Weighted universal Value-at-Risk Superadditivity for discrete distributions

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  • Alfred Muller

Abstract

The concept of weighted universal Value-at-Risk superadditivity (WUVS) was recently introduced by Chen et al. (2026) as a generalization of the question whether for some infinite mean distributions convex combinations of i.i.d. random variables can stochastically dominate the parent distribution. In this short note we prove that the property WUVS can basically never hold for discrete distributions except for the case of comonotonicity. This implies as a corollary that for discrete distributions with infinite mean it can also never hold that convex combinations of i.i.d. random variables can stochastically dominate the parent distribution. This settles an open problem mentioned in M\"uller (2025).

Suggested Citation

  • Alfred Muller, 2026. "Weighted universal Value-at-Risk Superadditivity for discrete distributions," Papers 2609.26398, arXiv.org.
    • Handle: RePEc:arx:papers:2609.26398
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    References listed on IDEAS

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    1. Chen, Yuyu & Embrechts, Paul & Wang, Ruodu, 2025. "Risk exchange under infinite-mean Pareto models," Insurance: Mathematics and Economics, Elsevier, vol. 124(C).
    2. Chen, Yuyu & Shneer, Seva, 2026. "Risk aggregation and stochastic dominance for a class of heavy-tailed distributions," ASTIN Bulletin, Cambridge University Press, vol. 56(1), pages 206-219, January.
    3. Yuyu Chen & Paul Embrechts & Ruodu Wang, 2025. "Technical Note—An Unexpected Stochastic Dominance: Pareto Distributions, Dependence, and Diversification," Operations Research, INFORMS, vol. 73(3), pages 1336-1344, May.
    4. Yuyu Chen & Paul Embrechts & Ruodu Wang, 2024. "Risk exchange under infinite-mean Pareto models," Papers 2403.20171, arXiv.org, revised Jun 2025.
    5. Alfred Muller, 2024. "Some remarks on the effect of risk sharing and diversification for infinite mean risks," Papers 2411.10139, arXiv.org, revised Mar 2025.
    6. Keyi Zeng & Zhenfeng Zou & Yuting Su & Taizhong Hu, 2025. "Further Developments on Stochastic Dominance for Convex Combinations of Infinite-Mean Random Variables," Papers 2511.00764, arXiv.org, revised Apr 2026.
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