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

Let $P$ be a Markov kernel on a measurable space $\text{X}$ and let $V:\text{X}\mathring{[}1,+\mathrm{\infty })$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels ${\text{Pc}}_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of ${\text{Pc}}_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the ${\text{Pc}}_k$-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 $\text{X}:=\mathbb{N}$.