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 $n geq 2$, there exist $n$ standard Cauchy random variables adding up to a constant $C$ if and only if $$|C|lerac{n,log (n-1)}{pi}.$$
Centers of probability measures without the mean / Giovanni Puccetti; Pietro Rigo; Bin Wang; Ruodu Wang. - In: JOURNAL OF THEORETICAL PROBABILITY. - ISSN 0894-9840. - STAMPA. - 32:(2019), pp. 1482-1501. [10.1007/s10959-018-0815-3]
Centers of probability measures without the mean
Pietro Rigo;
2019
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 $n geq 2$, there exist $n$ standard Cauchy random variables adding up to a constant $C$ if and only if $$|C|lerac{n,log (n-1)}{pi}.$$I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.