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Mat. Sb., 2011, Volume 202, Number 8, Pages 21–40 (Mi msb7717)  

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

Abstract: 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.

Keywords: rates of convergence in ergodic theorems, spectral measures, correlation coefficients, wide-sense stationary processes.

DOI: https://doi.org/10.4213/sm7717

Full text: PDF file (564 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2011, 202:8, 1105–1125

Bibliographic databases:

Document Type: Article
UDC: 517.987+519.214
MSC: Primary 28D05, 37A30; Secondary 37A50, 47A35, 60G10
Received: 20.03.2010 and 30.09.2010

Citation: 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

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. 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  mathnet  crossref  mathscinet  isi
    3. 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  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. 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  mathnet  crossref  mathscinet  isi
    6. 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  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    7. 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  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    8. 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  mathnet  crossref  elib
    9. 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  crossref  zmath  isi  scopus
    10. Kachurovskii A.G., Podvigin I.V., “Fejer Sums For Periodic Measures and the Von Neumann Ergodic Theorem”, Dokl. Math., 98:1 (2018), 344–347  crossref  mathscinet  zmath  isi  scopus
    11. Kachurovskii A.G., Podvigin I.V., “Fejer Sums and Fourier Coefficients of Periodic Measures”, Dokl. Math., 98:2 (2018), 464–467  crossref  mathscinet  zmath  isi  scopus
    12. K. I. Knizhov, I. V. Podvigin, “O skhodimosti integrala Luzina i ego analogov”, Sib. elektron. matem. izv., 16 (2019), 85–95  mathnet  crossref
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