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.
Citation:
A. G. Kachurovskii, V. V. Sedalishchev, “Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems”, Sb. Math., 202:8 (2011), 1105–1125
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\paper Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems
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Linking options:
https://www.mathnet.ru/eng/sm7717
https://doi.org/10.1070/SM2011v202n08ABEH004180
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This publication is cited in the following 17 articles:
Moacir Aloisio, Silas L. Carvalho, César R. de Oliveira, Edson Souza, “On spectral measures and convergence rates in von Neumann's Ergodic theorem”, Monatsh Math, 203:3 (2024), 543
A. G. Kachurovskii, I. V. Podvigin, A. J. Khakimbaev, “Uniform Convergence on Subspaces in von Neumann Ergodic
Theorem with Discrete Time”, Math. Notes, 113:5 (2023), 680–693
A. G. Kachurovskii, I. V. Podvigin, V. E. Todikov, “Uniform convergence on subspaces in von Neumann's ergodic theorem with continuous time”, Sib. elektron. matem. izv., 20:1 (2023), 183–206
Ben-Artzi J., Morisse B., “Uniform Convergence in Von Neumann'S Ergodic Theorem in the Absence of a Spectral Gap”, Ergod. Theory Dyn. Syst., 41:6 (2021), PII S0143385720000309, 1601–1611
A. G. Kachurovskii, M. N. Lapshtaev, A. Zh. Khakimbaev, “Ergodicheskaya teorema fon Neimana i summy Feiera zaryadov na okruzhnosti”, Sib. elektron. matem. izv., 17 (2020), 1313–1321
K. I. Knizhov, I. V. Podvigin, “O skhodimosti integrala Luzina i ego analogov”, Sib. elektron. matem. izv., 16 (2019), 85–95
A. G. Kachurovskii, K. I. Knizhov, “Deviations of Fejer sums and rates of convergence in the von Neumann ergodic theorem”, Dokl. Math., 97:3 (2018), 211–214
A. G. Kachurovskii, I. V. Podvigin, “Fejer sums for periodic measures and the von Neumann ergodic theorem”, Dokl. Math., 98:1 (2018), 344–347
A. G. Kachurovskii, I. V. Podvigin, “Fejer sums and Fourier coefficients of periodic measures”, Dokl. Math., 98:2 (2018), 464–467
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
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
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
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
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
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