This article is cited in 18 papers

Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems

A. G. Kachurovskii^a, V. V. Sedalishchev<sup>b

a</sup> Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
^b Novosibirsk State University, Mechanics and Mathematics Department

Abstract: The paper investigates estimates which relate two equivalent phenomena: the power-type rate of convergence in von Neumann's ergodic theorem and the power-type singularity at zero (with the same exponent) exhibited by the spectral measure of the function being averaged with respect to the corresponding dynamical system. The same rate of convergence is also estimated in terms of the rate of decrease of the correlation coefficients. Also, constants are found in analogous estimates for the power-type convergence in Birkhoff's ergodic theorem. All the results have exact analogues for wide-sense stationary stochastic processes.
Bibliography: 15 titles.

Keywords: rates of convergence in ergodic theorems, spectral measures, correlation coefficients, wide-sense stationary processes.

UDC: 517.987+519.214

MSC: Primary 28D05, 37A30; Secondary 37A50, 47A35, 60G10

Received: 20.03.2010 and 30.09.2010

DOI: 10.4213/sm7717

 English version: Sbornik: Mathematics, 2011, 202:8, 1105–1125

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