Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems. (English. Russian original) Zbl 1370.37014
Trans. Mosc. Math. Soc. 2016, 1-53 (2016); translation from Tr. Mosk. Mat. O.-va 77, No. 1, 1-66 (2016).
This article is a survey of results about the rate of convergence of ergodic averages in the theorems of Birkhoff and von Neumann. The estimates for the rate of convergence are expressed in terms of spectral measures and correlation coefficients. Let \((X,\mathcal B,\mu)\) be a probability space and \(T:X\to X\) be the corresponding dynamical map. Denote by \(S_n(\varphi)(x)\) the Birkhoff ergodic average for a map \(\varphi:X\to\mathbb R\) in \(L^1(X,\mu)\). If \(\varphi^\ast\) is the point-wise limit of \(S_n(\varphi)\) then to study the rate of convergence of the ergodic average one must analyze the decay of the sequence \(P_{n,\varepsilon}=\mu(\sup_{k\geq n}\{| S_n(\varphi)-\varphi^\ast|>\varepsilon\})\) for \(n\to\infty\). This is a problem of large deviations indeed. Rates for continuous-time dynamical systems and for operators are also considered. If \(L^0_2(X)\) is the subspace of \(L_2(X)\) which consists of maps with average zero then to obtain the mentioned rates, the authors analyze firstly the variance \(\| S_n(\varphi-\varphi^\ast)\|^2_2=\| S_n(\varphi)-\varphi^\ast\|\), i.e., the measure of deviation the in \(L_2\)-norm of \(S_n(\varphi)\) from the limit \(\varphi^\ast\). Thus bounds for the variance in terms of the spectral measures \(\sigma_{\varphi-\varphi^\ast}\) are obtained. These estimates are applied to estimate the rates convergence in the Birkhoff and von Neumann theorems.
Reviewer: Alejandro Mesón (La Plata)
MSC:
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
| 60G10 | Stationary stochastic processes |
Keywords:
convergence rates in ergodic theorems; correlation decay; large deviation decay; billiard; Anosov systemReferences:
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