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Rates of convergence of the Hastings and Metropolis algorithms. (English) Zbl 0854.60065

Summary: We apply recent results in Markov chain theory to Hastings and Metropolis algorithms with either independent or symmetric candidate distributions, and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution \(\pi\). In the independence case (in \(\mathbb{R}^k)\) these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of \(\pi\); in the symmetric case (in \(\mathbb{R}\) only) we show geometric convergence essentially occurs if and only if \(\pi\) has geometric tails. We also evaluate recently developed computable bounds on the rates of convergence in this context: examples show that these theoretical bounds can be inherently extremely conservative, although when the chain is stochastically monotone the bounds may well be effective.

MSC:

60J05 Discrete-time Markov processes on general state spaces
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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