Abstract:
Convergence almost everywhere of series ∑akξk is studied, where {ξk} is a wide-sense stationary sequence (or a quasi-stationary sequence). Sufficient conditions are obtained for convergence of the series, which are also necessary in the class of all sequences {ξk} having a given rate of decrease of the correlation function.
Analogous results are also valid for integrals of the type ∫∞1a(t)ξ(t)dt where ξ(t) is a wide-sense stationary process.
Bibliography: 12 titles.
\Bibitem{Gap75}
\by V.~F.~Gaposhkin
\paper Convergence of series connected with stationary sequences
\jour Math. USSR-Izv.
\yr 1975
\vol 9
\issue 6
\pages 1297--1321
\mathnet{http://mi.mathnet.ru/eng/im2096}
\crossref{https://doi.org/10.1070/IM1975v009n06ABEH001522}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=402880}
\zmath{https://zbmath.org/?q=an:0326.60038}
Linking options:
https://www.mathnet.ru/eng/im2096
https://doi.org/10.1070/IM1975v009n06ABEH001522
https://www.mathnet.ru/eng/im/v39/i6/p1366
This publication is cited in the following 14 articles:
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
Arkady Tempelman, “Randomized consistent statistical inference for random processes and fields”, Stat Inference Stoch Process, 25:3 (2022), 599
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
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, 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
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
N. A. Dzhulaǐ, A. G. Kachurovskiǐ, “Constants in the estimates of the rate of convergence in von Neumann's ergodic theorem with continuous time”, Siberian Math. J., 52:5 (2011), 824–835
A. G. Kachurovskii, V. V. Sedalishchev, “On the Constants in the Estimates of the Rate of Convergence in von Neumann's Ergodic Theorem”, Math. Notes, 87:5 (2010), 720–727
P. A. Yaskov, “A Generalization of the Menshov–Rademacher Theorem”, Math. Notes, 86:6 (2009), 861–872
Guy Cohen, Michael Lin, Characteristic Functions, Scattering Functions and Transfer Functions, 2009, 77
Guy Cohen, Michael Lin, “Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory”, Isr J Math, 148:1 (2005), 41
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
V. F. Gaposhkin, “Some Examples of the Problem of εε-Deviatations for Stationary Sequences”, Theory Probab Appl, 46:2 (2002), 341
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