Constants in the estimates of the convergence rate in the Birkhoff ergodic theorem with continuous time
type: Item
instance of: scholarly article
title: Constants in the estimates of the convergence rate in the Birkhoff ergodic theorem with continuous time
zbMATH Open document ID: 1260.37006
DOI: 10.1134/S0037446612050138
author: V. V. Sedalishchev
published in: Siberian Mathematical Journal
publication date: 2013-01-31T00:00:00Z
review text: 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 \(\phi(\alpha,t)=12\bigl(1+{\pi^2\over6}{1+2^{1-\alpha}\over(1-2^{1-\alpha})^3}\bigr)t^{-1}+16t^{-2}\).
reviewed by: Makoto Mori
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 37A10
zbMATH DE Number: 6133182
zbMATH Keywords: ergodic theorem
zbMATH Keywords: convergence rate
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6133182