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Long-term concentration of measure and cut-off

Author

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  • Barbour, A.D.
  • Brightwell, Graham
  • Luczak, Malwina

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.

Suggested Citation

  • Barbour, A.D. & Brightwell, Graham & Luczak, Malwina, 2022. "Long-term concentration of measure and cut-off," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 378-423.
    • Handle: RePEc:eee:spapps:v:152:y:2022:i:c:p:378-423
    DOI: 10.1016/j.spa.2022.05.004
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    References listed on IDEAS

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    1. Graham, Carl & Méléard, Sylvie, 1993. "Propagation of chaos for a fully connected loss network with alternate routing," Stochastic Processes and their Applications, Elsevier, vol. 44(1), pages 159-180, January.
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