The Kendall's Theorem and its Application to the Geometric Ergodicity of Markov Chains

TL;DR

This paper provides a precise quantitative version of Kendall's Theorem, enhancing understanding of renewal sequences and their convergence, which is crucial for analyzing the stability and convergence rates of Markov chains and MCMC methods.

Contribution

It introduces a sharp quantitative version of Kendall's Theorem, applying it to assess geometric ergodicity and convergence rates of Markov chains.

Findings

Established a sharp quantitative bound for Kendall's Theorem

Applied results to measure convergence rates of geometrically ergodic Markov chains

Provided estimates for convergence of MCMC estimators

Abstract

In this paper we prove a sharp quantitative version of the Kendall's Theorem. The Kendal Theorem states that under some mild conditions imposed on a probability distribution on positive integers (i.e. probabilistic sequence) one can prove convergence of its renewal sequence. Due to the well-known property - the first entrance last exit decomposition - such results are of interest in the stability theory of time homogeneous Markov chains. In particular the approach may be used to measure rates of convergence of geometrically ergodic Markov chains and consequently implies estimates on convergence of MCMC estimators.