  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.
