Empirical processes of iterated maps that contract on average

TL;DR

This paper investigates the weak convergence of empirical processes derived from a Markov chain formed by randomly iterating Lipschitz maps that contract on average, expanding understanding of their probabilistic behavior.

Contribution

It provides new results on the weak convergence of empirical processes for Markov chains with average contraction properties, a less explored area.

Findings

Established weak convergence under average contraction conditions

Extended empirical process theory to a new class of Markov chains

Provided conditions for convergence in distribution

Abstract

We consider a Markov chain obtained by random iterations of Lipschitz maps $T_i$ chosen with a probability $p_i(x)$ depending on the current position $x$. We assume this system has a property of "contraction on average", that is $\sum_i d(T_ix,T_iy)p_i(x) < \rho d(x,y)$ for some $\rho<1$. In the present note, we study the weak convergence of the empirical process associated to this Markov chain.