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Centers of probability measures without the mean

Author

Listed:
  • Giovanni Puccetti

    (University of Milano)

  • Pietro Rigo

    (University of Pavia)

  • Bin Wang

    (Chinese Academy of Sciences)

  • Ruodu Wang

    (University of Waterloo)

Abstract

In the recent years, the notion of mixability has been developed with applications to operations research, optimal transportation, and quantitative finance. An n-tuple of distributions is said to be jointly mixable if there exist n random variables following these distributions and adding up to a constant, called center, with probability one. When the n distributions are identical, we speak of complete mixability. If each distribution has finite mean, the center is obviously the sum of the means. In this paper, we investigate the set of centers of completely and jointly mixable distributions not having a finite mean. In addition to several results, we show the (possibly counterintuitive) fact that, for each $$n \ge 2$$ n ≥ 2 , there exist n standard Cauchy random variables adding up to a constant C if and only if $$\begin{aligned} |C|\le \frac{n\,\log (n-1)}{\pi }. \end{aligned}$$ | C | ≤ n log ( n - 1 ) π .

Suggested Citation

  • Giovanni Puccetti & Pietro Rigo & Bin Wang & Ruodu Wang, 2019. "Centers of probability measures without the mean," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1482-1501, September.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-018-0815-3
    DOI: 10.1007/s10959-018-0815-3
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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.
    2. E. G. Coffman & M. Yannakakis, 1984. "Permuting Elements Within Columns of a Matrix in Order to Minimize Maximum Row Sum," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 384-390, August.
    3. Carole Bernard & Ludger Rüschendorf & Steven Vanduffel, 2017. "Value-at-Risk Bounds With Variance Constraints," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 84(3), pages 923-959, September.
    4. Embrechts, Paul & Puccetti, Giovanni & Rüschendorf, Ludger, 2013. "Model uncertainty and VaR aggregation," Journal of Banking & Finance, Elsevier, vol. 37(8), pages 2750-2764.
    5. Paul Embrechts & Haiyan Liu & Ruodu Wang, 2017. "Quantile-Based Risk Sharing," Swiss Finance Institute Research Paper Series 17-54, Swiss Finance Institute.
    6. Wang, Bin & Wang, Ruodu, 2011. "The complete mixability and convex minimization problems with monotone marginal densities," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1344-1360, November.
    7. Lee, Woojoo & Ahn, Jae Youn, 2014. "On the multidimensional extension of countermonotonicity and its applications," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 68-79.
    8. Bernard, Carole & Jiang, Xiao & Wang, Ruodu, 2014. "Risk aggregation with dependence uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 93-108.
    9. Paul Embrechts & Giovanni Puccetti, 2006. "Bounds for Functions of Dependent Risks," Finance and Stochastics, Springer, vol. 10(3), pages 341-352, September.
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    Cited by:

    1. Pietro Rigo, 2020. "A Note on Duality Theorems in Mass Transportation," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2337-2350, December.
    2. Andrzej Komisarski & Jacques Labuschagne, 2024. "The Distributions of the Mean of Random Vectors with Fixed Marginal Distribution," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2121-2129, September.
    3. Takaaki Koike & Liyuan Lin & Ruodu Wang, 2022. "Joint mixability and notions of negative dependence," Papers 2204.11438, arXiv.org, revised Jan 2024.
    4. Yuyu Chen & Ruodu Wang, 2024. "Infinite-mean models in risk management: Discussions and recent advances," Papers 2408.08678, arXiv.org, revised Oct 2024.

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