Error bounds in a smooth metric for Brownian approximation of dynamical systems via Stein's method
Juho Leppänen, Yuto Nakajima, Yushi Nakano
TL;DR
The paper develops a Stein's method framework to quantify Brownian (diffusion) approximations in the functional central limit theorem for chaotic dynamical systems, under a functional correlation bound that yields an $O(N^{-1/2})$ error when variance grows linearly and correlations decay polynomially. It introduces test-function spaces ${\mathscr L},{\mathscr M},{\mathscr M}_0$ and constructs a Brownian Stein operator, establishing an abstract theorem that bounds $|\mu[ g(W_N) ] - \mathbf{E}[g(Z)]|$ for $g$ in ${\mathscr M}_0$. The theory is then applied to four dynamical-model families—Pikovsky maps, (non)autonomous and autonomous Liverani--Saussol--Vaienti maps, random Lasota--Yorke maps, and dispersing Sinai billiards—deriving explicit rates (often $O(N^{-1/2})$) under polynomial or exponential decay of correlations and positive limiting variance. This work extends Barbour’s diffusion-approximation approach to deterministic chaotic systems, offering new, explicit error bounds and broad applicability in nonuniformly expanding contexts, with potential implications for quantitative probabilistic analysis in complex dynamical settings.
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.