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

Let $\{X_n\}_{n\in\N}$ be a Markov chain on a measurable space $\X$ with transition kernel $P$ and let $V:\X\r[1,+\infty)$. The Markov kernel $P$ is here considered as a linear bounded operator on the weighted-supremum space $\cB_V$ associated with $V$. Then the combination of quasi-compactness arguments with precise analysis of eigen-elements of $P$ allows us to estimate the geometric rate of convergence $\rho_V(P)$ of $\{X_n\}_{n\in\N}$ to its invariant probability measure in operator norm on $\cB_V$. A general procedure to compute $\rho_V(P)$ for discrete Markov random walks with identically distributed bounded increments is specified.