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Centers of probability measures without the mean

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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 \(n \ge 2\), there exist n standard Cauchy random variables adding up to a constant C if and only if

$$\begin{aligned} |C|\le \frac{n\,\log (n-1)}{\pi }. \end{aligned}$$

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Authors and Affiliations

  1. Department of Economics, Management and Quantitative Methods, University of Milano, Milan, Italy

    Giovanni Puccetti

  2. Department of Mathematics, University of Pavia, Pavia, Italy

    Pietro Rigo

  3. Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Bin Wang

  4. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada

    Ruodu Wang

Authors
  1. Giovanni Puccetti
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  2. Pietro Rigo
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  3. Bin Wang
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  4. Ruodu Wang
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Correspondence to Giovanni Puccetti.

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Puccetti, G., Rigo, P., Wang, B. et al. Centers of probability measures without the mean. J Theor Probab 32, 1482–1501 (2019). https://doi.org/10.1007/s10959-018-0815-3

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  • DOI: https://doi.org/10.1007/s10959-018-0815-3

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