Centers of Probability Measures Without the Mean

Abstract

In the recent years, the notion of mixability has been developed with applications to optimal transportation, quantitative finance and operations research. 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 ≥ 2, there exist n standard Cauchy random variables adding up to a constant C if and only if |C| ≤ n log(n − 1) π .MSC2000 subject classification: Primary 60E05, Secondary 90B30.(1)Even if apparently innocuous, the proof (or at least our proof) of such a result is quite involved.