Long-term concentration of measure and cut-off

A.D. Barbour, Graham Brightwell and Malwina Luczak

Stochastic Processes and their Applications, 2022, vol. 152, issue C, 378-423

Abstract: We present new concentration of measure inequalities for Markov chains, generalising results for chains that are contracting in Wasserstein distance. These are particularly suited to establishing the cut-off phenomenon for suitable chains. We apply our discrete-time inequality to the well-studied Bernoulli–Laplace model of diffusion, and give a probabilistic proof of cut-off, recovering and improving the bounds of Diaconis and Shahshahani. We also extend the notion of cut-off to chains with an infinite state space, and illustrate this in a second example, of a two-host model of disease in continuous time. We give a third example, giving concentration results for the supermarket model, illustrating the full generality and power of our results.

Keywords: Markov chains; Concentration of measure; Coupling; Cut-off (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:152:y:2022:i:c:p:378-423

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DOI: 10.1016/j.spa.2022.05.004

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