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On aggregation sets and lower-convex sets

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  • Mao, Tiantian
  • Wang, Ruodu

Abstract

It has been a challenge to characterize the set of all possible sums of random variables with given marginal distributions, referred to as an aggregation set in this paper. We study the aggregation set via its connection to the corresponding lower-convex set, which is the set of all sums of random variables that are smaller than the respective marginal distributions in convex order. Theoretical properties of the two sets are discussed, assuming that all marginal distributions have finite mean. In particular, an aggregation set is always a subset of its corresponding lower-convex set, and the two sets are identical in the asymptotic sense after scaling. We also show that a lower-convex set is identical to the set of comonotonic sums with the same marginal constraint. The main theoretical results contribute to the field of multivariate distributions with fixed margins.

Suggested Citation

  • Mao, Tiantian & Wang, Ruodu, 2015. "On aggregation sets and lower-convex sets," Journal of Multivariate Analysis, Elsevier, vol. 138(C), pages 170-181.
    • Handle: RePEc:eee:jmvana:v:138:y:2015:i:c:p:170-181
    DOI: 10.1016/j.jmva.2014.12.001
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    References listed on IDEAS

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    1. Ruodu Wang & Liang Peng & Jingping Yang, 2013. "Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities," Finance and Stochastics, Springer, vol. 17(2), pages 395-417, April.
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    Cited by:

    1. Ruodu Wang & Yunran Wei & Gordon E. Willmot, 2020. "Characterization, Robustness, and Aggregation of Signed Choquet Integrals," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 993-1015, August.
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    3. Xia Han & Qiuqi Wang & Ruodu Wang & Jianming Xia, 2021. "Cash-subadditive risk measures without quasi-convexity," Papers 2110.12198, arXiv.org, revised May 2024.

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