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 Sibirsk. Mat. Zh., 2014, Volume 55, Number 2, Pages 412–426 (Mi smj2543)

Interrelation between the convergence rates in von Neumann's and Birkhoff's ergodic theorems

V. V. Sedalishchev

Novosibirsk State University, Novosibirsk, Russia

Abstract: In the $L_p$ spaces, $1<p<\infty$, we prove some inequalities for discrete and continuous times that make it possible to obtain the convergence rate in Birkhoff's theorem in the presence of bounds on the convergence rate in von Neumann's ergodic theorem belonging to a sufficiently large rate range. The exact operator analogs of these inequalities for contraction semigroups in $L_p$ are given. These results also have the obvious exact analogs in the class of wide-sense stationary stochastic processes.

Keywords: von Neumann's ergodic theorem, Birkhoff's ergodic theorem, convergence rate of ergodic averages, wide-sense stationary stochastic process, contraction semigroups in $L_p$.

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English version:
Siberian Mathematical Journal, 2014, 55:2, 336–348

Bibliographic databases:

UDC: 517.987+519.214

Citation: V. V. Sedalishchev, “Interrelation between the convergence rates in von Neumann's and Birkhoff's ergodic theorems”, Sibirsk. Mat. Zh., 55:2 (2014), 412–426; Siberian Math. J., 55:2 (2014), 336–348

Citation in format AMSBIB
\Bibitem{Sed14} \by V.~V.~Sedalishchev \paper Interrelation between the convergence rates in von Neumann's and Birkhoff's ergodic theorems \jour Sibirsk. Mat. Zh. \yr 2014 \vol 55 \issue 2 \pages 412--426 \mathnet{http://mi.mathnet.ru/smj2543} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3237344} \transl \jour Siberian Math. J. \yr 2014 \vol 55 \issue 2 \pages 336--348 \crossref{https://doi.org/10.1134/S0037446614020165} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000335167300016} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84899687263} 

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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. I. V. Podvigin, “On the rate of convergence in the individual ergodic theorem for the action of a semigroup”, Siberian Adv. Math., 26:2 (2016), 139–151
2. 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
3. I. V. Podvigin, “Estimates for correlation in dynamical systems: from Hölder continuous functions to general observables”, Siberian Adv. Math., 28:3 (2018), 187–206
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