zbMATH — the first resource for mathematics

Comparison of cutoffs between lazy walks and Markovian semigroups. (English) Zbl 1288.60088
Summary: 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.

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J27 Continuous-time Markov processes on discrete state spaces
47D07 Markov semigroups and applications to diffusion processes
PDF BibTeX Cite
Full Text: DOI Euclid arXiv
[1] Aldous, D. and Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. Appl. Math. 8 , 69-97. · Zbl 0631.60065
[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 , 69-74. · Zbl 1133.60342
[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 , 26-78. · Zbl 1190.60007
[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 , 293-321. · Zbl 1297.60049
[7] Diaconis, P. (1988). Group Representations in Probability and Statistics . Institute of Mathematical Statistics, Hayward, CA. · Zbl 0695.60012
[8] Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. USA. 93 , 1659-1664. · Zbl 0849.60070
[9] Diaconis, P. and Saloff-Coste, L. (2006). Separation cut-offs for birth and death chains. Ann. Appl. Prob. 16 , 2098-2122. · Zbl 1127.60081
[10] Ding, J., Lubetzky, E. and Peres, Y. (2010). Total variation cutoff in birth-and-death chains. Prob. Theory Relat. Fields 146 , 61-85. · Zbl 1190.60005
[11] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis . Cambridge University Press. · Zbl 0704.15002
[12] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times . American Mathematical Society, Providence, RI. · Zbl 1160.60001
[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. 301-413. · Zbl 0885.60061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.