  In this paper we prove a sharp quantitative version of the Kendall's Theorem.
The Kendal Theorem states that under some mild conditions imposed on a
probability distribution on positive integers (i.e. probabilistic sequence) one
can prove convergence of its renewal sequence. Due to the well-known property -
the first entrance last exit decomposition - such results are of interest in
the stability theory of time homogeneous Markov chains. In particular the
approach may be used to measure rates of convergence of geometrically ergodic
Markov chains and consequently implies estimates on convergence of MCMC
estimators.
