The rate of convergence in ergodic theorems.

*(English. Russian original)*Zbl 0880.60024
Russ. Math. Surv. 51, No. 4, 653-703 (1996); translation from Usp. Mat. Nauk 51, No. 4, 73-124 (1996).

Consider an automorphism \(T\) of a probability space \((\Omega, \lambda)\) and a function \(f \in L_1(\Omega)\). The well-known counterexamples show that in the pointwise ergodic theorem (E.T.) it is impossible to give estimates of the convergence rates for averages \(\frac{1}{n} \sum_{k \leq n} f\circ T^k\) depending on \(f\) alone, i.e. being uniform on the automorphism group. In contrast, this survey concentrates on positive results including some new results of the author. In particular it is stressed that the here considered oscillations of averages admit estimates involving \(f\) alone, moreover, depending only on \( |f |_r\). The paper is organized as follows: introduction, 3 chapters, 3 appendices and the bibliography comprising 46 references. The first chapter deals with the rate of convergence in the pointwise E.T. Here many interesting problems are discussed, e.g. spectral measures and asymptotic behaviour of variances for certain sums, decay of probability of an \(\varepsilon\)-deviation etc. Chapter 2 is devoted to the analysis of oscillation of averages in the pointwise E.T. (e.g. \(\varepsilon\)-fluctuations, p-variation). Chapter 3 provides results on the rate of convergence and oscillations in other E.T. Appendix 1 gives an interpretation of certain claims in terms of non-standard analysis. Appendix 2 (S. A. Utev) contains estimates of large deviations of a random number of fluctuations of averages for independent summands. Appendix 3 concentrates at fluctuations of bounded martingales.

Reviewer: A.V.Bulinskij (Moskva)

##### MSC:

60F15 | Strong limit theorems |