Error bounds in a smooth metric for Brownian approximation of dynamical systems via Stein's method

TL;DR

This paper extends Stein's method to chaotic dynamical systems, providing explicit error bounds in the functional central limit theorem for systems with polynomial decay of correlations, applicable to complex systems like Sinai billiards.

Contribution

It adapts Stein's diffusion approximation method to chaotic systems, establishing error bounds under correlation decay conditions, and applies to various complex dynamical models.

Findings

Error bounds of order O(N^{-1/2}) for systems with polynomial decay

Application to interval maps with neutral fixed points

Application to Sinai billiards

Abstract

We adapt Stein's method of diffusion approximations, developed by Barbour, to the study of chaotic dynamical systems. We establish an error bound in the functional central limit theorem with respect to an integral probability metric of smooth test functions under a functional correlation decay bound. For systems with a sufficiently fast polynomial rate of correlation decay, the error bound is of order $O(N^{-1/2})$, under an additional condition on the linear growth of variance. Applications include a family of interval maps with neutral fixed points and unbounded derivatives, and two-dimensional dispersing Sinai billiards.