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

Let $\{X_n{\}}_{n\in \mathbb{N}}$ be a Markov chain on a measurable space $\text{X}$ with transition kernel $P$ and let $V:\text{X}\mathring{[}1,+\mathrm{\infty })$. The Markov kernel $P$ is here considered as a linear bounded operator on the weighted-supremum space ${\text{cB}}_V$ 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 $\{X_n{\}}_{n\in \mathbb{N}}$ to its invariant probability measure in operator norm on ${\text{cB}}_V$. A general procedure to compute ${\rho }_V(P)$ for discrete Markov random walks with identically distributed bounded increments is specified.