Separation cut-offs for birth and death chains

Details Separation cut-offs for birth and death chains Separation cut-offs for birth and death chains

2007-02-14

arxiv.org/abs/math/0702411v1

Annals of Applied Probability 2006, Vol. 16, No. 4, 2098-2122

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051606000000501 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051606000000501

This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cut-off. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there is a cut-off if and only if the product of spectral gap and mixing time tends to infinity. We establish this for arbitrary birth and death chains in continuous time when the convergence is measured in separation and the chains all start at 0.

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