Constants in the estimates of the convergence rate in the Birkhoff ergodic theorem with continuous time
For the case of continuous time, under the assumption of the power law of the von Neumann ergodic theorem, one gets the estimate of convergence rate of the Birkhoff ergodic theorem. Let \(T_t\) be a semi-flow on a probability space \((\Omega,\mathfrak S,\lambda)\). Denote the time average of the flow by \[ A_tf(\omega)={1\over t}\int_0^tf(T^\tau\omega)\,d\tau, \] and by \(f^*\) the limit in \(L_2(\Omega)\) of \(A_tf\). The main result of this article is: { Theorem 2. } Suppose that \(||A_tf-f^*||_2^2\leq B(t+1)^{-\alpha}\) for all \(t\), with \(B\) a positive constant and \(0<\alpha\leq2\). Then \[ \lambda\{\sup_{s\geq t}|A_sf-f^*|\geq\varepsilon\}<\phi(\alpha,t)B\varepsilon^{-2} \] for all \(\varepsilon>0\) and \(t\geq2\). Moreover, depending on \(\alpha\) 1. if \(\alpha\in(0,1)\), then \(\phi(\alpha,t)={3\cdot2^\alpha\over1-2^{-\alpha}} \bigl(1+{1\over(1-2^{\alpha-1\over2})^2}\bigr)t^{-\alpha}+16t^{-2}\), 2. if \(\alpha=1\), then \(\phi(\alpha,t)=12((1+\log_2t)^2+3)t^{-1}+16t^{-2}\), 3. if \(\alpha\in(1,2]\), then \(\phi(\alpha,t)=12\bigl(1+{\pi^2\over6}{1+2^{1-\alpha}\over(1-2^{1-\alpha})^3}\bigr)t^{-1}+16t^{-2}\).
scientific article; zbMATH DE number 6133182
scientific article