The Halász–Székely barycenter. Issue 4 (13th November 2022)

• Record Type: Journal Article
• Authors: Bochi, Jairo; Iommi, Godofredo; Ponce, Mario
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 65:Issue 4(2022)
• Page Start: 881
• Page End: 911
• Publication Date: 2022-11-13
• Subjects: symmetric mean -- barycenter -- ergodic theorem; 26E60 -- 26D15 -- 15A15 -- 37A30 -- 60F15; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091522000372
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of non-negative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We study the analytic properties of this Halász–Székely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of non-negative 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 converge to the Halász–Székely barycenter of the corresponding distribution.