Approximating Markov chains and V-geometric ergodicity via weak perturbation theory

TL;DR

This paper establishes explicit links between the ergodic properties of a Markov kernel and its finite-rank approximations, providing bounds on convergence rates and invariant measures using spectral perturbation theory.

Contribution

It introduces a method to relate the $V$-geometric ergodicity of a Markov kernel to that of its finite-rank approximations via weak perturbation theory, with explicit bounds.

Findings

Derived bounds for total variation distance between invariant measures.

Provided a spectral procedure for estimating convergence rates.

Applied the method to truncations of discrete Markov kernels.

Abstract

Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels $\Pc_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $\Pc_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the $\Pc_k$-invariant positive measure. The proofs are based on the Keller-Liverani perturbation theorem which requires an accurate control of the essential spectral radius of $P$ on usual weighted supremum spaces. Such computable bounds are derived in terms of standard drift conditions. Our spectral procedure to estimate both the convergence rate and the invariant probability measure of $P$ is applied to truncation of discrete Markov kernels on $\X:=\N$.