Asymptotic Variance and Convergence Rates of Nearly-Periodic Markov Chain Monte Carlo Algorithms

This article considers nearly-periodic Markov chains that may have excellent functional estimation properties but poor distributional convergence rate. It shows how simple modifications of the chain (involving using a random number of iterations) can greatly improve the distributional convergence of the chain. Various theoretical results about convergence rates of the modified chains are proven. A number of examples, including a transdimensional Markov chain Monte Carlo example, a card-shuffling example, and several antithetic Metropolis algorithms, are considered.