The rate of convergence in ergodic theorems

Abstract

Contents Introduction 0.1. Notation 0.2. On uniform estimates 0.3. Brief description of new results Chapter I. Rate of convergence in the pointwise ergodic theorem § 1. Growth of the dispersion 1.1. Spectral measures and power-function growth of dispersion 1.2. Proof of Theorem 3 1.3. Correlation coefficients and dispersion growth § 2. Decay of the probability of an ε-deviation 2.1. The case of independent 2.2. Decay of and growth of 2.3. On the rate of approximation of by functions cohomologous to zero 2.4. Proofs of Theorems 11 and 12 § 3. On the law of the iterated logarithm 3.1. The growth of and the law of the iterated logarithm § 4. On uniform convergence 4.1. The fastest uniform convergence 4.2. Two criteria for weak mixing Chapter II. Oscillation of averages in the pointwise ergodic theorem § 5. Crossings of an interval § 6. ε-fluctuations § 7. p-variation Chapter III. Rate of convergence and oscillations in other ergodic theorems § 8. Rate of convergence § 9. Oscillations Appendix 1. Interpretation in terms of non-standard analysis A1.1. The theory of internal sets A1.2. Elementary analogues of ergodic theorems Appendix 2. Estimates of large deviations of the random number of fluctuations of averages for independent terms A2.1. Formulation of the basic result A2.2. Outline of the proof of Theorem 33 Appendix 3. Fluctuations of bounded martingales Bibliography