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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. 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.
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    Cited by:

    1. Takaaki Koike & Liyuan Lin & Ruodu Wang, 2022. "Joint mixability and notions of negative dependence," Papers 2204.11438, arXiv.org, revised Jan 2024.
    2. Pietro Rigo, 2020. "A Note on Duality Theorems in Mass Transportation," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2337-2350, December.

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