Exponential inequalities for unbounded functions of geometrically ergodic Markov chains. Applications to quantitative error bounds for regenerative Metropolis algorithms

TL;DR

This paper develops exponential inequalities for unbounded functions of geometrically ergodic Markov chains, enabling the derivation of confidence intervals and quantitative error bounds for regenerative Metropolis algorithms in MCMC.

Contribution

It introduces new concentration inequalities for unbounded functions of Markov chains, extending existing results and applying them to improve error bounds in MCMC methods.

Findings

Derived exponential inequalities for unbounded functions

Provided confidence intervals for MCMC algorithms

Quantitative error bounds for regenerative Metropolis

Abstract

The aim of this note is to investigate the concentration properties of unbounded functions of geometrically ergodic Markov chains. We derive concentration properties of centered functions with respect to the square of the Lyapunov's function in the drift condition satisfied by the Markov chain. We apply the new exponential inequalities to derive confidence intervals for MCMC algorithms. Quantitative error bounds are providing for the regenerative Metropolis algorithm of [5].