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Comparison of Cutoffs Between Lazy Walks and Markovian Semigroups

  • Guan-Yu Chen (a1) and Laurent Saloff-Coste (a2)

Abstract

We make a connection between the continuous time and lazy discrete time Markov chains through the comparison of cutoffs and mixing time in total variation distance. For illustration, we consider finite birth and death chains and provide a criterion on cutoffs using eigenvalues of the transition matrix.

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Copyright

Corresponding author

Postal address: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan. Email address: gychen@math.nctu.edu.tw
∗∗ Postal address: Malott Hall, Department of Mathematics, Cornell University, Ithaca, NY 14853-4201. Email address: lsc@math.cornell.edu

References

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[1] Aldous, D. and Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. Appl. Math. 8, 6997.
[2] Aldous, D. and Fill, J. Reversible Markov Chains and Random Walks on Graphs. Available at http://www. stat.berkeley.edu/users/aldous/RWG/book.html.
[3] Brown, M. and Shao, Y.-S. (1987). Identifying coefficients in the spectral representation for first passage time distributions. Prob. Eng. Inf. Sci. 1, 6974.
[4] Chen, G.-Y. (2006). The cutoff phenomenon for finite Markov chains. Doctoral Thesis, Cornell University.
[5] Chen, G.-Y. and Saloff-Coste, L. (2008). The cutoff phenomenon for ergodic Markov processes. Electron. J. Prob. 13, 2678.
[6] Chen, G.-Y. and Saloff-Coste, L. (2013). On the mixing time and spectral gap for birth and death chains. ALEA Lat. Amer. J. Prob. Math. Statist. 10, 293321.
[7] Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics, Hayward, CA.
[8] Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. USA. 93, 16591664.
[9] Diaconis, P. and Saloff-Coste, L. (2006). Separation cut-offs for birth and death chains. Ann. Appl. Prob. 16, 20982122.
[10] Ding, J., Lubetzky, E. and Peres, Y. (2010). Total variation cutoff in birth-and-death chains. Prob. Theory Relat. Fields 146, 6185.
[11] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Cambridge University Press.
[12] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. American Mathematical Society, Providence, RI.
[13] Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (St-Flour, 1996; Lecture Notes Math. 1665), Springer, Berlin, pp. 301413.

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Comparison of Cutoffs Between Lazy Walks and Markovian Semigroups

  • Guan-Yu Chen (a1) and Laurent Saloff-Coste (a2)

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