On the rate of convergence of ergodic averages for functions of Gordin space
Автор: Podvigin Ivan V.
Журнал: Владикавказский математический журнал @vmj-ru
Статья в выпуске: 2 т.26, 2024 года.
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For an automorphisms with non-zero Kolmogorov-Sinai entropy, a new class of L2-functions called the Gordin space is considered. This space is the linear span of Gordin classes constructed by some automorphism-invariant filtration of σ-algebras Fn. A function from the Gordin class is an orthogonal projection with respect to the operator I-E(⋅|Fn) of some Fm-measurable function. After Gordin's work on the use of the martingale method to prove the central limit theorem, this construction was developed in the works of Voln\'{y}. In this review article we consider this construction in ergodic theory. It is shown that the rate of convergence of ergodic averages in the L2 norm for functions from the Gordin space is simply calculated and is O(1n√). It is also shown that the Gordin space is a dense set of the first Baire category in L2(Ω,F,μ)⊖L2(Ω,Π(T,F),μ), where Π(T,F) is the Pinsker σ-algebra.
Rates of convergence in ergodic theorems, filtration, martingale method
Короткий адрес: https://sciup.org/143182664
IDR: 143182664 | DOI: 10.46698/w0408-5668-5674-e
Список литературы On the rate of convergence of ergodic averages for functions of Gordin space
- Gordin, M. I. The Central Limit Theorem for Stationary Processes, Doklady Akademii Nauk SSSR, 1969, vol. 188, no. 4, pp. 739-741 (in Russian).
- Volny, D. Approximating Martingales and the Central Limit Theorem for Strictly Stationary Processes, Stochastic Processes and Their Applications, 1993, vol. 44, no. 1, pp. 41-74. DOI: 10.1016/0304-4149(93)90037-5.
- Gordin, M. I. A Note on the Martingale Method of Proving the Central Limit Theorem for Stationary Sequences, Journal of Mathematical Sciences, 2006, vol. 133, no. 3, pp. 1277-1281. DOI: 10.1007/s10958-006-0036-7.
- Wu, W. B. and Woodroofe, M. Martingale Approximations for Sums of Stationary Processes, The Annals of Probability, 2004, vol. 32, no. 2, pp. 1674-1690. DOI: 10.1214/009117904000000351.
- Cuny, C. and Fan, A. H. Study of Almost Everywhere Convergence of Series by Mean of Martingale Methods, Stochastic Processes and their Applications, 2017, vol. 127, no. 8, pp. 2725-2750. DOI: 10.1016/j.spa.2016.12.006.
- Lesigne, E. and Volny, D. Large Deviations for Martingales, Stochastic Processes and Their Applications, 2001, vol. 96, no. 1, pp. 143-159. DOI: 10.1016/S0304-4149(01)00112-0.
- Melbourne, I. and Nicol, M. Large Deviations for Nonuniformly Hyperbolic Systems, Transactions of the American Mathematical Society, 2008, vol. 360, no. 12, pp. 6661-6676. DOI: 10.1090/S0002-9947-08-04520-0.
- Melbourne, I. Large and Moderate Deviations for Slowly Mixing Dynamical Systems, Proceedings of the American Mathematical Society, 2009, vol. 137, no. 5, pp. 1735-1741. DOI: 10.1090/S0002-9939-08-09751-7.
- Chazottes, J.-R., Cuny, C., Dedecker, J., Fan, X. and Lemler, S. Limit Theorems and Inequalities via Martingale Methods, ESAIM: Proceedings, 2014, vol. 44, pp. 177-196. DOI: 10.1051/proc/201444012.
- Browder, F. E. On the Iteration of Transformations in Noncompact Minimal Dynamical Systems, Proceedings of the American Mathematical Society, 1958, vol. 9, no. 5, pp. 773-780. DOI: 10.2307/2033085.
- Badea, C. and Devys, O. Rochberg's Abstract Coboundary Theorem Revisited, Complex Analysis and Operator Theory, 2022, vol. 16, no. 8, article number 115. DOI: 10.1007/s11785-022-01293-w.
- Derriennic, Y. and Lin, M. Fractional Poisson Equations and Ergodic Theorems for Fractional Coboundaries, Israel Journal of Mathematics, 2001, vol. 123, no. 1, pp. 93-130. DOI: 10.1007/BF02784121.
- Glasner, E. Ergodic Theory via Joinings, Mathematical Surveys and Monographs, vol. 101, American Mathematical Society, 2003, 384 p.
- Assani, I. Wiener Wintner Ergodic Theorems, Singapore, Word Scientific, 2003, 216 p.
- Kachurovskii, A. G. and Podvigin, I. V. Estimates of the Rates of Convergence in the von Neumann and Birkhoff Ergodic Theorem, Transactions of the Moscow Mathematical Society, 2016, vol. 77, pp. 1-53. DOI: 10.1090/mosc/256.
- Kachurovskii, A. G., Podvigin, I. V. and Khakimbaev, A. J. Uniform Convergence on Subspaces in von Neumann Ergodic Theorem with Discrete Time, Mathematical Notes, 2023, vol. 113, no. 5, pp. 680-693. DOI: 10.1134/S0001434623050073.
- Cuny, C. Pointwise Ergodic Theorems with Rate with Applications to Limit Theorems for Stationary Processes, Stochastics and Dynamics, 2011, vol. 11, no. 1, pp. 135-155. DOI: 10.1142/ S0219493711003206.
- Adler, R. L. Invariant and Reducing Subalgebras of Measure Preserving Transformations, Transactions of the American Mathematical Society, 1964, vol. 110, no. 2, pp. 350-360. DOI: 10.1090/S0002-9947-1964-0156941-9.
- Sinai, Ya. G. On a Weak Isomorphism of Transformations with Invariant Measure, Matematicheskii Sbornik. Novaya Seriya, 1964, vol. 63(105), no 1, pp. 23-42 (in Russian).
- Katznelson, Y. Ergodic Automorphisms of Tn are Bernoulli Shifts, Israel Journal of Mathematics, 1971, vol. 10, pp. 186-195. DOI: 10.1007/BF02771569.
- Kornfeld I. P., Sinai, Ya. G. and Fomin, S. V. Ergodic Theory, Berlin-Heidelberg-New York, Springer, 1982, 486 p.
- Volny, D. Martingale Decompositions of Stationary Processes, Yokohama Mathematical Journal, 1987, vol. 35, pp. 113-121.
- Podvigin, I. V. On Possible Estimates of the Rate of Pointwise Convergence in the Birkhoff Ergodic Theorem, Siberian Mathematical Journal, 2022, vol. 63, no. 2, pp. 316-325. DOI: 10.1134/S0037446622020094.
