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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 aprobability distribution on positive integers (i.e. probabilistic sequence) onecan prove convergence of its renewal sequence. Due to the well-known property -the first entrance last exit decomposition - such results are of interest inthe stability theory of time homogeneous Markov chains. In particular theapproach may be used to measure rates of convergence of geometrically ergodicMarkov chains and consequently implies estimates on convergence of MCMCestimators.

The Kendall's Theorem and its Application to the Geometric Ergodicity of Markov Chains