CLTs and Asymptotic Variance of Time-Sampled Markov Chains

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For a Markov transition kernel P and a probability distribution μ on nonnegative integers, a time-sampled Markov chain evolves according to the transition kernel \(P_{\mu} = \sum_k \mu(k)P^k.\) In this note we obtain CLT conditions for time-sampled Markov chains and derive a spectral formula for the asymptotic variance. Using these results we compare efficiency of Barker’s and Metropolis algorithms in terms of asymptotic variance.