Computable 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 $r\_ess(P\_|{\ell }^2(\pi ))\le \alpha \_0$. Effective bounds on the convergence rate can be provided from a truncation procedure.

in Journal of Applied Probability, Applied Probability Trust, 2016. arXiv admin note: substantial text overlap with arXiv:1503.02206

References in corpus (4)

• Renewal theory and computable convergence rates for geometrically ergodic Markov chains
• Approximating Markov chains and V-geometric ergodicity via weak perturbation theory
• Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts
• Additional material on bounds of ${\ell }^2$-spectral gap for discrete Markov chains with band transition matrices