Stein’s method of normal approximation for dynamical systems

Abstract

We present an adaptation of Stein's method of normal approximation to the study of both discrete-and continuous-time dynamical systems. We obtain new correlation-decay conditions on dynamical systems for a multivariate central limit theorem augmented by a rate of convergence. We then present a scheme for checking these conditions in actual examples. The principal contribution of our paper is the method, which yields a convergence rate essentially with the same amount of work as the central limit theorem, together with a multiplicative constant that can be computed directly from the assumptions.

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

“…A similar result (with different constants) was recently proved in [5], but there a direct scheme for checking (A2) was implemented. Here we illustrate that (A1) and (A2) -as well as the bound in (4) -are immediate consequences of Theorem 2.4.…”

“…The following theorem was proved in [5], where an adaptation of Stein's method [14] to the study of dynamical systems was developed:…”

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

“…The purpose of this paper is to show that the answer is positive in the case of the intermittent Pomeau-Manneville maps of a suitable parameter range. We demonstrate how these maps satisfy a certain functional correlation bound (Theorem 1.1) from which the conditions of [10,15] readily follow. As results, we obtain two versions of the multivariate CLT with speed in this setting of Pomeau-Manneville maps.…”

“…After conditions on A have been derived, they can be translated to conditions on h by using the explicit solution to the equation (12). Indeed, D k h ∞ < ∞ for 1 ≤ k ≤ 3 implies that [5,10]. These steps lead to the following result.…”

“…We say that it satisfies the central limit theorem (CLT), if the normalized Birkhoff sums N −1/2 ∑ N−1 k=0 f • T k converge in distribution to a d-dimensional Gaussian random variable. In [10], certain correlationdecay conditions were formulated by using Stein's method of normal approximation. Once the system is shown to satisfy these conditions, multivariate CLT augmented by a rate of convergence follows immediately.…”