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 nonnegative 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. AMS subject classification : 60J10; 47B07