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Variance bounding Markov chains. (English) Zbl 1142.60047

Summary: We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, variance bounding is equivalent to the existence of usual central limit theorems for all \(L^{2}\) functionals. Also, variance bounding (unlike geometric ergodicity) is preserved under the Peskun order. We close with some applications to Metropolis-Hastings algorithms.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
65C40 Numerical analysis or methods applied to Markov chains
47A10 Spectrum, resolvent
60J22 Computational methods in Markov chains
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