Additional material on bounds of $\ell^2$-spectral gap for discrete Markov chains with band transition matrices

TL;DR

This paper analyzes the spectral gap and convergence rates of discrete Markov chains with band transition matrices, providing new bounds and conditions for geometric ergodicity, with applications to algorithms like Metropolis-Hastings.

Contribution

It introduces new criteria based on spectral radius and transition probabilities to estimate convergence rates of Markov chains with band matrices, including reversible cases.

Findings

Derived bounds for the essential spectral radius of Markov chains.

Established conditions for spectral gap (SG2) based on transition probabilities.

Applied results to estimate convergence rates in Metropolis-Hastings algorithms.

Abstract

We analyse the $\ell^2(\pi)$-convergence rate of irreducible and aperiodic Markov chains with $N$-band transition probability matrix $P$ and with invariant distribution $\pi$. This analysis is heavily based on: first the study of the essential spectral radius $r\_{ess}(P\_{|\ell^2(\pi)})$ of $P\_{|\ell^2(\pi)}$ derived from Hennion's quasi-compactness criteria; second the connection between the spectral gap property (SG$\_2$) of $P$ on $\ell^2(\pi)$ and the $V$-geometric ergodicity of $P$. Specifically, (SG$\_2$) is shown to hold under the condition \[\alpha\_0 := \sum\_{{m}=-N}^N \limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\, 1. \] Moreover $r\_{ess}(P\_{|\ell^2(\pi)}) \leq \alpha\_0$. Simple conditions on asymptotic properties of $P$ and of its invariant probability distribution $\pi$ to ensure that $\alpha\_0\textless{}1$ are given. In particular this allows us to obtain estimates of the $\ell^2(\pi)$-geometric convergence rate of random walks with bounded increments. The specific case of reversible $P$ is also addressed. Numerical bounds on the convergence rate can be provided via a truncation procedure. This is illustrated on the Metropolis-Hastings algorithm.