Abstract
The hitting and mixing times are two fundamental quantities associated with Markov chains. In Peres and Sousi [PS15] and Oliveira [Oli12], the authors show that the mixing times and “worst-case” hitting times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on general state spaces that satisfy the strong Feller property. Finally, we show that this asymptotic equivalence can be used to find bounds on the mixing times of a large class of Markov chains used in MCMC, such as typical Gibbs samplers and Metropolis–Hastings chains, even though they usually do not satisfy the strong Feller property.
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The authors owe a debt of gratitude to the anoymous referee for an extremely detailed report, as well as many insightful suggestions to improve both the mathematical content and the presentation of our paper.
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Anderson, R.M., Duanmu, H. & Smith, A. Mixing times and hitting times for general Markov processes. Isr. J. Math. 259, 759–834 (2024). https://doi.org/10.1007/s11856-023-2555-z
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DOI: https://doi.org/10.1007/s11856-023-2555-z