Spectral analysis of Markov kernels and application to the convergence rate of discrete random walks

Abstract

Let {X_n}_n∈N be a Markov chain on a measurable space X with transition kernel P, and let V : X → [1, +∞). The Markov kernel P is here considered as a linear bounded operator on the weighted-supremum space B_V associated with V. Then the combination of quasicompactness arguments with precise analysis of eigenelements of P allows us to estimate the geometric rate of convergence ρ_V(P) of {X_n}_n∈N to its invariant probability measure in operator norm on B_V. A general procedure to compute ρ_V(P) for discrete Markov random walks with identically distributed bounded increments is specified.