This article is cited in 12 scientific papers (total in 12 papers)
Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems
A. G. Kachurovskiia, V. V. Sedalishchevb
a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b Novosibirsk State University, Mechanics and Mathematics Department
The paper investigates estimates which relate two equivalent phenomena: the power-type rate of convergence in von Neumann's ergodic theorem and the power-type singularity at zero (with the same exponent) exhibited by the spectral measure of the function being averaged with respect to the corresponding dynamical system. The same rate of convergence is also estimated in terms of the rate of decrease of the correlation coefficients. Also, constants are found in analogous estimates for the power-type convergence in Birkhoff's ergodic theorem. All the results have exact analogues for wide-sense stationary stochastic processes.
Bibliography: 15 titles.
rates of convergence in ergodic theorems, spectral measures, correlation coefficients, wide-sense stationary processes.
PDF file (564 kB)
Sbornik: Mathematics, 2011, 202:8, 1105–1125
MSC: Primary 28D05, 37A30; Secondary 37A50, 47A35, 60G10
Received: 20.03.2010 and 30.09.2010
A. G. Kachurovskii, V. V. Sedalishchev, “Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems”, Mat. Sb., 202:8 (2011), 21–40; Sb. Math., 202:8 (2011), 1105–1125
Citation in format AMSBIB
\by A.~G.~Kachurovskii, V.~V.~Sedalishchev
\paper Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems
\jour Mat. Sb.
\jour Sb. Math.
Citing articles on Google Scholar:
Related articles on Google Scholar:
This publication is cited in the following articles:
A. G. Kachurovskii, V. V. Sedalishchev, “On the Constants in the Estimates of the Rate of Convergence in the Birkhoff Ergodic Theorem”, Math. Notes, 91:4 (2012), 582–587
V. V. Sedalishchev, “Constants in the estimates of the convergence rate in the Birkhoff ergodic theorem with continuous time”, Siberian Math. J., 53:5 (2012), 882–888
Kachurovskii A.G., Podvigin I.V., “Rates of convergence in ergodic theorems for certain billiards and Anosov diffeomorphisms”, Dokl. Math., 88:1 (2013), 385–387
A. G. Kachurovskii, I. V. Podvigin, “Large Deviations and the Rate of Convergence in the Birkhoff Ergodic Theorem”, Math. Notes, 94:4 (2013), 524–531
V. V. Sedalishchev, “Interrelation between the convergence rates in von Neumann's and Birkhoff's ergodic theorems”, Siberian Math. J., 55:2 (2014), 336–348
A. G. Kachurovskii, I. V. Podvigin, “Rate of convergence in ergodic theorems for the planar periodic Lorentz gas”, Dokl. Math., 89:2 (2014), 139–142
A. G. Kachurovskii, I. V. Podvigin, “Correlations, large deviations, and rates of convergence in ergodic theorems for characteristic functions”, Dokl. Math., 91:2 (2015), 204–207
A. G. Kachurovskii, I. V. Podvigin, “Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems”, Trans. Moscow Math. Soc., 77 (2016), 1–53
Kachurovskii A.G., Knizhov K.I., “Deviations of Fejer Sums and Rates of Convergence in the Von Neumann Ergodic Theorem”, Dokl. Math., 97:3 (2018), 211–214
Kachurovskii A.G., Podvigin I.V., “Fejer Sums For Periodic Measures and the Von Neumann Ergodic Theorem”, Dokl. Math., 98:1 (2018), 344–347
Kachurovskii A.G., Podvigin I.V., “Fejer Sums and Fourier Coefficients of Periodic Measures”, Dokl. Math., 98:2 (2018), 464–467
K. I. Knizhov, I. V. Podvigin, “O skhodimosti integrala Luzina i ego analogov”, Sib. elektron. matem. izv., 16 (2019), 85–95
|Number of views:|