Journal
ANNALS OF APPLIED PROBABILITY
Volume 16, Issue 4, Pages 2098-2122Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/105051606000000501
Keywords
ergodic Markov chains; birth and death chains; mixing time; strong stationary time
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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 cutoff 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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