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 (Formula presented.), there exist n standard Cauchy random variables adding up to a constant C if and only if (Formula presented.)

Centers of probability measures without the mean / G. Puccetti, P. Rigo, B. Wang, R. Wang. - In: JOURNAL OF THEORETICAL PROBABILITY. - ISSN 0894-9840. - 32:3(2019 Sep 01), pp. 1482-1501. [10.1007/s10959-018-0815-3]

### Centers of probability measures without the mean

#### 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 (Formula presented.), there exist n standard Cauchy random variables adding up to a constant C if and only if (Formula presented.)
##### Scheda breve Scheda completa Scheda completa (DC)
Cauchy distribution; Complete mixability; Joint mixability; Multivariate dependence;
Settore SECS-S/06 - Metodi mat. dell'economia e Scienze Attuariali e Finanziarie
Settore SECS-S/01 - Statistica
Settore MAT/06 - Probabilita' e Statistica Matematica
1-set-2019
10-feb-2018
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2434/552321`