A note on the Kawada-Ito theorem


Mustafayev H.

STATISTICS & PROBABILITY LETTERS, vol.181, 2022 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 181
  • Publication Date: 2022
  • Doi Number: 10.1016/j.spl.2021.109261
  • Journal Name: STATISTICS & PROBABILITY LETTERS
  • Journal Indexes: Science Citation Index Expanded, Scopus, Academic Search Premier, MathSciNet, zbMATH, DIALNET
  • Keywords: Mean ergodic theorem, Locally compact group, Probability measure, Convergence

Abstract

A probability measure mu on a locally compact group G is said to be adapted if the support of mu generates a dense subgroup of G. A classical Kawada-Ito theorem asserts that if mu is an adapted measure on a compact metrizable group G, then the sequence of probability measures {1/n Sigma(n=1)(k=0) mu(k)}(n=1)(infinity) weak* converges to the Haar measure on G. In this note, we present a new proof of Kawada-Ito theorem. Also, we show that metrizability condition in the Kawada-Ito theorem can be removed. Some applications are also given. (C) 2021 Elsevier B.V. All rights reserved.