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Total variation cutoff in birth-and-death chains
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  • Published: 07 November 2008

Total variation cutoff in birth-and-death chains

  • Jian Ding1,
  • Eyal Lubetzky2 &
  • Yuval Peres2 

Probability Theory and Related Fields volume 146, Article number: 61 (2010) Cite this article

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Abstract

The cutoff phenomenon describes a case where a Markov chain exhibits a sharp transition in its convergence to stationarity. Diaconis [Proc Natl Acad Sci USA 93(4):1659–1664, 1996] surveyed this phenomenon, and asked how one could recognize its occurrence in families of finite ergodic Markov chains. Peres [American Institute of Mathematics (AIM) Research Workshop, Palo Alto. http://www.aimath.org/WWN/mixingtimes, 2004] noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing-time and spectral-gap tends to infinity, and conjectured that in many settings, this condition should also be sufficient. Diaconis and Saloff-Coste [Ann Appl Probab 16(4):2098–2122, 2006] verified this conjecture for continuous-time birth-and-death chains, started at an endpoint, with convergence measured in separation. It is natural to ask whether the conjecture holds for these chains in the more widely used total-variation distance. In this work, we confirm the above conjecture for all continuous-time or lazy discrete-time birth-and-death chains, with convergence measured via total-variation distance. Namely, if the product of the mixing-time and spectral-gap tends to infinity, the chains exhibit cutoff at the maximal hitting time of the stationary distribution median, with a window of at most the geometric mean between the relaxation-time and mixing-time. In addition, we show that for any lazy (or continuous-time) birth-and-death chain with stationary distribution π, the separation 1 − p t(x, y)/π(y) is maximized when x, y are the endpoints. Together with the above results, this implies that total-variation cutoff is equivalent to separation cutoff in any family of such chains.

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Authors and Affiliations

  1. Department of Statistics, UC Berkeley, Berkeley, CA, 94720, USA

    Jian Ding

  2. Microsoft Research, One Microsoft Way, Redmond, WA, 98052-6399, USA

    Eyal Lubetzky & Yuval Peres

Authors
  1. Jian Ding
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  2. Eyal Lubetzky
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  3. Yuval Peres
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Corresponding author

Correspondence to Eyal Lubetzky.

Additional information

Research of J. Ding and Y. Peres was supported in part by NSF grant DMS-0605166.

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Ding, J., Lubetzky, E. & Peres, Y. Total variation cutoff in birth-and-death chains. Probab. Theory Relat. Fields 146, 61 (2010). https://doi.org/10.1007/s00440-008-0185-3

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  • Received: 04 April 2008

  • Revised: 23 September 2008

  • Published: 07 November 2008

  • DOI: https://doi.org/10.1007/s00440-008-0185-3

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Mathematics Subject Classification (2000)

  • 60B10
  • 60J05
  • 60J27
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