Geometric ergodicity for families of homogeneous Markov chains

TL;DR

This paper establishes uniform exponential deviation bounds for families of homogeneous Markov chains and applies these results to nonparametric estimation of ergodic diffusion processes.

Contribution

It provides sufficient conditions for geometric ergodicity uniformly over parameter families and applies these to nonparametric estimation.

Findings

Derived nonasymptotic exponential bounds for ergodic theorem deviations.

Identified conditions ensuring uniform geometric ergodicity.

Applied results to nonparametric estimation of ergodic diffusions.

Abstract

In this paper we find nonasymptotic exponential upper bounds for the deviation in the ergodic theorem for families of homogeneous Markov processes. We find some sufficient conditions for geometric ergodicity uniformly over a parametric family. We apply this property to the nonasymptotic nonparametric estimation problem for ergodic diffusion processes.