Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard

Abstract

We show how Rio's method [Probab. Theory Related Fields 104 (1996) 255-282] can be adapted to establish a rate of convergence in 1 √ n in the multidimensional central limit theorem for some stationary processes in the sense of the Kantorovich metric. We give two applications of this general result: in the case of the Knudsen gas and in the case of the Sinai billiard.

Stein’s method of normal approximation for dynamical systems

“…We mention a further benefit of the above features of Stein's method: the method turns out to be sufficiently flexible to be implemented in the non-stationary setup in which the random vectors f • T k are replaced by f • T k • · · · • T 1 with a rather arbitrary sequence of maps T i , i ≥ 1, which generally do not preserve a common measure. This appears to be a big difference compared to Pène's adaptation [40] of Rio's method. (Further differences will be discussed in Section 7.3.)…”

“…In the present context of dynamical systems (i.e., measure-preserving transformations), the question has, of course, been long studied in a number of landmark papers, e.g., [16,24,25,29,[38][39][40], using different methods, hypotheses, and metrics. To our knowledge, however, Pène's important article [40] is the only one treating the multidimensional case d > 1, building on a method due to Rio [45]. We note that while a multivariate central limit theorem can be reduced to the univariate case, bounds on the rate of convergence do not pass from d = 1 to higher dimensions.…”

“…This paper grew out of interest in investigating how it can be adapted to the setup of dynamical systems. Stein's method stands out among other methods of normal approximation known to us at least in two regards: Firstly, it does 1 not resort to the use of characteristic functions (Fourier transforms) of the distributions, even under the surface (in contrast with, say, the martingale central limit theorem or the method used in [40]). Rather, it allows us to turn the problem of normal convergence into a set of problems on correlation decay.…”

Functional correlation decay and multivariate normal approximation for non-uniformly expanding maps

“…The second result in this vein will be established by invoking the main result of [15] which is again applicable by virtue of Theorem 1.1.…”

A Note on the Finite-Dimensional Distributions of Dispersing Billiard Processes

“…The result is due to Pène [11], assuming piecewise Hölder continuous observables. The above version covers also dynamically Hölder continuous observables.…”