Spectral measures and rates of convergence in the ergodic theorem (Q1377971)

Let \((\Omega, {\mathcal B}, P)\) be a probability space and \(T\) its automorphism. For \(f\in L_1 (\Omega)\) let \(A_nf(x)= n^{-1} \sum^{n-1}_{k=0} f(T^kx)\) and \(f^*(x)= \lim_{n\to \infty} A_n f(x)\). A rate of convergence to zero of \(p_n^\varepsilon =P(| A_nf-f^*| \geq \varepsilon)\) and \(P^\varepsilon_n =P(\sup_{j\geq n} | A_j f-f^* |\geq \varepsilon)\) is determined. It is shown that this rate depends on the concentration of the spectral measure \(\sigma_f\) of \(f\) at the vicinity of 0. One of the results is as follows: If \(\sigma_f([-\delta, \delta]) =O(\delta^\alpha)\) as \(\delta\to 0\) for some \(\alpha\in(0,1]\), then \(P_n^\varepsilon =O (n^{-\alpha})\) for \(f\in L_\infty\) and \(P_n^\varepsilon =O(n^{-\alpha/2})\) for \(f\in L_2\).