Computable bounds of ℓ2{\ell}^2-spectral gap for discrete Markov chains with band transition matrices

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 $r\_{ess}(P\_{|\ell^2(\pi)}) \leq \alpha\_0$. Effective bounds on the convergence rate can be provided from a truncation procedure.Comment: in Journal of Applied Probability, Applied Probability Trust, 2016. arXiv admin note: substantial text overlap with arXiv:1503.0220

Additional material on bounds of ℓ2\ell^2-spectral gap for discrete Markov chains with band transition matrices

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