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This article presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster&#x2013;Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof. 1) It relies entirely on functional-analytic non-probabilistic arguments. 2) It makes explicit the connection between a Foster&#x2013;Lyapunov function and Poincar&#x00E9; inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the nonreversible case is also presented.

On the Lyapunov Foster Criterion and Poincaré Inequality for Reversible Markov Chains