The Hal\'asz-Sz\'ekely Barycenter

TL;DR

This paper introduces the Halász-Székely barycenter, a new concept of barycenter linked to symmetric means of nonnegative real numbers, and explores its properties, inequalities, and convergence behavior in dynamical systems.

Contribution

It defines the Halász-Székely barycenter, analyzes its properties, and proves an ergodic theorem connecting symmetric means with this barycenter.

Findings

Established inequalities relating symmetric means and barycenters.

Proved an ergodic theorem for convergence of symmetric means.

Connected symmetric means with the Halász-Székely barycenter in dynamical systems.

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.