Frontiers of Mathematics in China >

2014 , Vol. 9 >Issue 4: 881 - 898

DOI: https://doi.org/10.1007/s11464-014-0380-8

RESEARCH ARTICLE

Explicit criteria on separation cutoff for birth and death chains

  • Yonghua MAO ,
  • Yuhui ZHANG
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  • School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received date: 03 Mar 2014

Accepted date: 03 Apr 2014

Published date: 20 Aug 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

The criteria on separation cutoff for birth and death chains were obtained by Diaconis and Saloff-Coste in 2006. These criteria are involving all eigenvalues. In this paper, we obtain the explicit criterion, which depends only on the birth and death rates. Furthermore, we present two ways to estimate moments of the fastest strong stationary time and then give another but equivalent criterion explicitly.

Key words： Separation cutoff; birth and death chain; hitting time; fastest strong stationary time (FSST); eigenvalue; stochastic monotonicity; duality; boundary theory

Cite this article

Yonghua MAO , Yuhui ZHANG . Explicit criteria on separation cutoff for birth and death chains[J]. Frontiers of Mathematics in China, 2014 , 9(4) : 881 -898 . DOI: 10.1007/s11464-014-0380-8

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Aldous D, Diaconis P. Strong uniform times and finite random walks. Adv in Appl Math, 1987, 8(1): 69-97

DOI

2
Aldous D, Fill J A. Reversible Markov Chains and Random Walks on Graphs. www.berkeley.edu/users/aldous/book.html, 2003

3
Billingsley P. Probability and Measures. 3rd ed. New York: Wiley, 1995

4
Chen M F. From Markov Chains to Non-Equilibrium Particle Systems. 2nd ed. Singapore: World Scientific, 2004

5
Chen M F. Speed of stability for birth-death processes. Front Math China, 2010, 5(3): 379-515

DOI

6
Diaconis P. The cutoff phenomenon in finite Markov chains. Proc Natl Acad Sci USA, 1996, 93(4): 1659-1664

DOI

7
Diaconis P, Fill J A. Strong stationary times via a new form of duality. Ann Probab, 1990, 18: 1483-1522

DOI

8
Diaconis P, Saloff-Coste L. What do we know about the Metropolis algorithm. J Comput System Sci, 1998, 57: 20-36

DOI

9
Diaconis P, Saloff-Coste L. Separation cut-offs for birth and death chains. Ann Appl Probab, 2006, 16(4): 2098-2122

DOI

10
Feller W. An Introduction to Probability Theory and Its Applications, II. 2nd ed. New York: Wiley, 1971

11
Fill J A. Strong stationary duality for continuous-time Markov chains, Part I: theory. J Theoret Probab, 1992, 5(1): 45-69

DOI

12
Gong Y, Mao Y H, Zhang C. Hitting time distribution for a denumerable birth-death process. J Theoret Probab, 2012, 25: 950-980

DOI

13
Levin D A, Peres Y, Wilmer E L. Markov Chains and Mixing Times.Providence: Amer Math Sos, 2008

14
Mao Y H. Algebraic convergence for discrete-time ergodic Markov chains. Science in China (Series A), 2003, 46(5): 621-630

DOI

15
Mao Y H. Ergodic degrees for continuous-time Markov chains. Sci China Ser A, 2004, 47(2): 161-174

DOI

16
Mao Y H. The eigentime identity for ergodic Markov chains. J Appl Probab, 2004, 41(4): 1071-1080

DOI

17
Mao Y H. Convergence rates for reversible Markov chains without the assumption of nonnegative definite matrices. Sci China Math, 2010, 53(8): 1979-1988

DOI

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