INEQUALITIES FOR ESTIMATING CONVERGENCE RATES IN ERGODIC THEOREMS

Abstract

Mean ergodic theorem convergence rate inequalities are obtained. The existence of such inequalities is implied by the equivalence between power-function convergence rate in this theorem and the presence of the power singularity (with the same exponent) of averaging function spectral measure at zero which is related with corresponding dynamical system. The same convergence rate was also estimated via correlation coefficients. Important for possible applications particular cases when correlation coefficients decay rate is power and exponential were considered apart. Estimates for pointwise ergodic theorem convergence rate were obtained for the case when mean ergodic theorem convergence rate is known. All results of the paper have their exact analogues for stationary in the wide sense stochastic processes.