# Centers of probability measures without the mean

## TL;DR

This paper explores the concept of centers in probability measures lacking finite means, revealing that certain sums of Cauchy distributions can be constant within specific bounds, extending mixability theory.

## Contribution

It characterizes the set of centers for completely and jointly mixable distributions without finite means, including a surprising result for sums of Cauchy variables.

## Key findings

- Existence of centers for sums of Cauchy distributions within specific bounds.
- Extension of mixability concepts to distributions without finite mean.
- Counterintuitive bounds for sum constants of Cauchy random variables.

## 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 $n \geq 2$, there exist $n$ standard Cauchy random variables adding up to a constant $C$ if and only if $$|C|\le\frac{n\,\log (n-1)}{\pi}.$$