Total variation cutoff in birth-and-death chains
The authors investigate the cutoff phenomenon for sequences of Markov chains where these chains exhibit sharp transitions in their convergence to their stationary distributions. It was observed recently by \textit{Y. Peres} [``Sharp thresholds for mixing times'', in: American Institute of Mathematics (AIM) Research Workshop, Palo Alto; \url{http://www.aimath.org/WWN/mixingtimes} (2004)] that a necessary condition for cutoff phenomena for sequences of reversible chains is that the products of mixing-times and spectral gaps tend to infinity. Moreover, it was shown recently by \textit{P. Diaconis} and \textit{L. Saloff-Coste} [Ann. Appl. Probab. 16, No.~4, 2098--2122 (2006; Zbl 1127.60081)] that this condition is also sufficient for continuous-time birth-and-death chains starting in end points when convergence is measured in separation. In the paper under review it is shown that the condition is also sufficient with respect to total variation distance for certain chains, namely for continuous-time birth-and-death chains as well as for socalled lazy discrete time birth-and-death chains where \(P(x,x)\geq1/2\) holds for all states \(x\). The main ingredient for the proof is a a sharp estimate for the difference for the mixing times \(t_{\text{mix}}(\varepsilon)-t_{\text{mix}}(1-\varepsilon)\) for \(\varepsilon\in]0,1/2[\) in terms of the product of the spectral gap and \(t_{\text{mix}}(1/4)\).
sufficient condition for cutoff-phenomenon
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