Geodesic
Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems
Sbornik. Mathematics, Tome 202 (2011) no. 8, pp. 1105-1125 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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, wide-sense stationary processes.
Mots-clés : correlation coefficients 
@article{SM_2011_202_8_a1,
     author = {A. G. Kachurovskii and V. V. Sedalishchev},
     title = {Constants in estimates for the rates of convergence in von {Neumann's} and {Birkhoff's} ergodic theorems},
     journal = {Sbornik. Mathematics},
     pages = {1105--1125},
     year = {2011},
     volume = {202},
     number = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2011_202_8_a1/}
}
TY  - JOUR
AU  - A. G. Kachurovskii
AU  - V. V. Sedalishchev
TI  - Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems
JO  - Sbornik. Mathematics
PY  - 2011
SP  - 1105
EP  - 1125
VL  - 202
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/SM_2011_202_8_a1/
LA  - en
ID  - SM_2011_202_8_a1
ER  - 
%0 Journal Article
%A A. G. Kachurovskii
%A V. V. Sedalishchev
%T Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems
%J Sbornik. Mathematics
%D 2011
%P 1105-1125
%V 202
%N 8
%U http://geodesic.mathdoc.fr/item/SM_2011_202_8_a1/
%G en
%F SM_2011_202_8_a1
A. G. Kachurovskii; V. V. Sedalishchev. Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems. Sbornik. Mathematics, Tome 202 (2011) no. 8, pp. 1105-1125. http://geodesic.mathdoc.fr/item/SM_2011_202_8_a1/

[1] I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai, Ergodic theory, Grundlehren Math. Wiss., 245, Springer-Verlag, New York–Heidelberg–Berlin, 1982 | MR | MR | Zbl | Zbl

[2] A. G. Kachurovskii, “The rate of convergence in ergodic theorems”, Russian Math. Surveys, 51:4 (1996), 653–703 | DOI | MR | Zbl

[3] 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 | DOI | MR | Zbl

[4] 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, 87:5–6 (2010), 720–727 | DOI | MR

[5] N. K. Bary, A treatise on trigonometric series, Macmillan, New York, 1964 | MR | MR | Zbl

[6] J. von Neumann, “Proof of the quasi-ergodic hypothesis”, Proc. Nat. Acad. Sci. U.S.A., 18:1 (1932), 70–82 | DOI | Zbl

[7] I. A. Ibragimov, Yu. V. Linnik, Independent and stationary sequences of random variables, Grundlehren Math. Wiss., 245, Wolters-Noordhoff Publ., Groningen, 1971 | MR | MR | Zbl | Zbl

[8] Ya. G. Sinai, Dynamical systems II. Ergodic theory with applications to dynamical systems and statistical mechanics, Encyclopaedia Math. Sci., 2, Springer-Verlag, Berlin, 1989 | MR | MR | Zbl | Zbl

[9] I. Assani, M. Lin, “On the one-sided ergodic Hilbert transform”, Ergodic theory and related fields (Chapel Hill, NC, USA, 2004–2006), Contemp. Math., 430, Amer. Math. Soc., Providence, RI, 2007, 21–39 | MR | Zbl

[10] V. F. Gaposhkin, “Decrease rate of the probabilities of $\varepsilon$-deviations for the means of stationary processes”, Math. Notes, 64:3 (1998), 316–321 | DOI | MR | Zbl

[11] V. F. Gapoškin, “Convergence of series connected with stationary sequences”, Math. USSR-Izv., 9:6 (1975), 1297–1321 | DOI | MR | Zbl | Zbl

[12] V. P. Leonov, “On the dispersion of time-dependent means of a stationary stochastic process”, Theory Probab. Appl., 6 (1961), 87–93 | DOI | MR | Zbl

[13] R. E. Edwards, Fourier series. A modern introduction, v. 1, 2, Grad. Texts in Math., 64, 85, Springer-Verlag, New York–Heidelberg–Berlin, 1979, 1982 | MR | MR | MR | Zbl | Zbl

[14] A. N. Shiryaev, Probability, Grad. Texts in Math., 95, Springer-Verlag, New York, 1996 | MR | MR | Zbl | Zbl

[15] V. F. Gaposhkin, “Criteria for the strong law of large numbers for some classes of second-order stationary processes and homogeneous random fields”, Theory Probab. Appl., 22:2 (1978), 286–310 | DOI | MR | Zbl