On convergence rate in ergodic theorem for some statistically averaging sequences in $\mathbb{R}$

Abstract

In this work we consider two types of averaging of unitary representation of the group $\mathbb{R}$ constructed by some sequences of probability measures on $\mathbb{R}.$ The first sequence of measures generalizes the uniform distribution. The densities of the measures in this sequence are convolutions of finitely many indicators of segments. The second sequence is defined by the exponential decay of Fourier transform. For such averagings we obtain the estimates for the convergence rate in the norm depending on the singularities of spectral measure of the unitary representation in a neighbourhood of zero and of the asymptotics of sequence of Fourier transforms of averaging probabilistic measures. At the same time, the maximal possible rates are powers with the exponents $m>1$ and the exponential rate, respectively, and this is significantly better than the maximal convergence rate in the classical von Neumann ergodic theorem.