The Halász–Székely barycenter

Abstract We introduce a notion of barycenter probability measure related to the symmetric mean collection non-negative real numbers. Our definition is inspired by work Halász and Székely, who in 1976 proved law large numbers for means. study analytic properties this Halász–Székely barycenter. establish fundamental inequalities that relate list with uniformly supported on these points. As consequence, we go an ergodic theorem stating means sequence dynamical observations converge corresponding distribution.