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

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 \[α\_0 := \sum\_{{m}=-N}^N \limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\, 1. \] Moreover $r\_ess(P\_|{\ell }^2(\pi ))\le \alpha \_0$. Simple conditions on asymptotic properties of $P$ and of its invariant probability distribution $\pi$ to ensure that $α\_0\textless{}1$ 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.

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• Computable bounds of ${\ell }^2$-spectral gap for discrete Markov chains with band transition matrices