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On the rate of convergence in von Neumann's ergodic theorem with continuous time
Sbornik. Mathematics, Tome 201 (2010) no. 4, pp. 493-500 Cet article a éte moissonné depuis la source Math-Net.Ru

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The rate of convergence in von Neumann's mean ergodic theorem is studied for continuous time. The condition that the rate of convergence of the ergodic averages be of power-law type is shown to be equivalent to requiring that the spectral measure of the corresponding dynamical system have a power-type singularity at 0. This forces the estimates for the convergence rate in the above ergodic theorem to be necessarily spectral. All the results obtained have obvious exact analogues for wide-sense stationary processes. Bibliography: 7 titles.
Keywords: von Neumann's mean ergodic theorem; rate of convergence of ergodic averages; spectral measures of a dynamical system; wide-sense stationary stochastic processes. 
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A. G. Kachurovskii; A. V. Reshetenko. On the rate of convergence in von Neumann's ergodic theorem with continuous time. Sbornik. Mathematics, Tome 201 (2010) no. 4, pp. 493-500. http://geodesic.mathdoc.fr/item/SM_2010_201_4_a1/

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