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 Mat. Zametki, 1998, Volume 64, Issue 3, Pages 366–372 (Mi mz1406)

Decrease rate of the probabilities of $\varepsilon$-deviations for the means of stationary processes

V. F. Gaposhkin

Moscow State University of Railway Communications

Abstract: The asymptotic behavior as $n\to\infty$ of the normed sums $\sigma_n=n^{-1}\sum_{k=0}^{n-1}X_k$ for a stationary process $X=(X_n, n\in\mathbb Z)$ is studied. For a fixed $\varepsilon>0$, upper estimates for $\mathsf P(\sup_{k\ge n} |\sigma_k|\ge\varepsilon)$ as $n\to\infty$ are obtained.

DOI: https://doi.org/10.4213/mzm1406

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English version:
Mathematical Notes, 1998, 64:3, 316–321

Bibliographic databases:

UDC: 519

Citation: V. F. Gaposhkin, “Decrease rate of the probabilities of $\varepsilon$-deviations for the means of stationary processes”, Mat. Zametki, 64:3 (1998), 366–372; Math. Notes, 64:3 (1998), 316–321

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Gaposhkin, VF, “Some examples of the problem of epsilon-deviations for stationary sequences”, Theory of Probability and Its Applications, 46:2 (2001), 341
2. V. F. Gaposhkin, “Estimates of the Entropy of the Set of Means for Some Classes of Stationary and Quasistationary Sequences”, Math. Notes, 78:1 (2005), 47–52
3. V. F. Gaposhkin, “Exact Estimates of the Metric Entropy of the Averages for Some Classes of Stationary Sequences”, Theory Probab. Appl., 53:1 (2009), 37–58
4. A. G. Kachurovskii, A. V. Reshetenko, “On the rate of convergence in von Neumann's ergodic theorem with continuous time”, Sb. Math., 201:4 (2010), 493–500
5. 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
6. Kachurovskii A.G., Sedalischev V.V., “Neravenstva, pozvolyayuschie otsenivat skorosti skhodimosti v ergodicheskikh teoremakh”, Vestnik Kemerovskogo gosudarstvennogo universiteta, 2011, no. 3-1, 250–254
7. 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
8. 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
9. Gomilko A., Haase M., Tomilov Yu., “Bernstein Functions and Rates in Mean Ergodic Theorems for Operator Semigroups”, J. Anal. Math., 118 (2012), 545–576
10. 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
11. 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
12. Kachurovskii A.G., Podvigin I.V., “Fejer Sums For Periodic Measures and the Von Neumann Ergodic Theorem”, Dokl. Math., 98:1 (2018), 344–347
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