The Hal\'asz-Sz\'ekely Barycenter
Probability
2023-03-28 v1 Dynamical Systems
Functional Analysis
Abstract
We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of nonnegative real numbers. Our definition is inspired by the work of Hal\'asz and Sz\'ekely, who in 1976 proved a law of large numbers for symmetric means. We study analytic properties of this Hal\'asz-Sz\'ekely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of nonnegative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converges to the Hal\'asz-Sz\'ekely barycenter of the corresponding distribution.
Cite
@article{arxiv.2103.05182,
title = {The Hal\'asz-Sz\'ekely Barycenter},
author = {Jairo Bochi and Godofredo Iommi and Mario Ponce},
journal= {arXiv preprint arXiv:2103.05182},
year = {2023}
}