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

Let P be a Markov kernel on a measurable space X and let V:X→[1,+∞). This paper provides explicit connections between the V-geometric ergodicity of P and that of finite-rank non-negative sub-Markov kernels P̂k approximating P. A special attention is paid to obtain an efficient way to specify the convergence rate for P from that of P̂k and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the P-invariant probability measure and the P̂k-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.