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Biometrika 1996 83(1):95-110; doi:10.1093/biomet/83.1.95
© 1996 by Biometrika Trust
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Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms

G. O. ROBERTS and R. L. TWEEDIE

Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge Cambridge CB2 1SB, U.K.
Department of Statistics, Colorado State University Fort Collins, Colorado 80523, U.S.A.

We develop results on geometric ergodicity of Markov chains and apply these and other recent results in Markov chain theory to multidimensional Hastings and Metropolis algorithms. For those based on random walk candidate distributions, we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution {pi}. By phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly-used statistical forms. From these results we develop central limit theorems for the Metropolis algorithm. Converse results, showing non-geometric convergence rates for chains where the rejection rate is not bounded away from unity, are also given; these show that the negative-definiteness property is not redundant.

Key Words: Geometric ergodicity • Gibbs sampling Hastings algorithm • Irreducible Markov process • Markov chain Monte Carlo • Metropolis algorithm • Posterior distribution


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