Wasserstein convergence rates in the invariance principle for sequential dynamical systems

TL;DR

This paper investigates how quickly the Wasserstein distance converges in the invariance principle for non-stationary sequential dynamical systems, extending techniques from stationary cases to a broader class of systems.

Contribution

It introduces modified methods to analyze Wasserstein convergence rates in non-stationary dynamical systems, applicable to various classes including sequential transformations and noisy expanding maps.

Findings

Established convergence rates for Wasserstein distance in non-stationary systems

Extended techniques from stationary to non-stationary dynamical systems

Applicable to a wide class of systems including noisy and piecewise expanding maps

Abstract

In this paper, we consider the convergence rate with respect to the Wasserstein distance in the invariance principle for sequential dynamical systems. We utilize and modify the techniques previously employed for stationary sequences to address our non-stationary case. Under certain assumptions, we can apply our result to a large class of dynamical systems, including sequential $\beta_n$-transformations, piecewise uniformly expanding maps with additive noise in one-dimensional and multidimensional case, and so on.