On the mixing time and spectral gap for birth and death chains

For birth and death chains, we derive bounds on the spectral gap and mixing time in terms of birth and death rates. Together with the results of Ding et al. in 2010, this provides a criterion for the existence of a cutoff in terms of the birth and death rates. A variety of illustrative examples are treated.

[1]  B. Muckenhoupt Hardy's inequality with weights , 1972 .

[2]  D. Aldous Random walks on finite groups and rapidly mixing markov chains , 1983 .

[3]  Mark Brown,et al.  Identifying Coefficients in the Spectral Representation for First Passage Time Distributions , 1987, Probability in the Engineering and Informational Sciences.

[4]  P. Diaconis Group representations in probability and statistics , 1988 .

[5]  P. Diaconis,et al.  Geometric Bounds for Eigenvalues of Markov Chains , 1991 .

[6]  P. Diaconis,et al.  COMPARISON THEOREMS FOR REVERSIBLE MARKOV CHAINS , 1993 .

[7]  P. Diaconis,et al.  Comparison Techniques for Random Walk on Finite Groups , 1993 .

[8]  Persi Diaconis,et al.  What Do We Know about the Metropolis Algorithm? , 1998, J. Comput. Syst. Sci..

[9]  L. Miclo An example of application of discrete Hardy''s inequalities , 1999 .

[10]  Guan-Yu Chen,et al.  The cutoff phenomenon for finite Markov Chains , 2006 .

[11]  Persi Diaconis,et al.  Separation cut-offs for birth and death chains , 2006, math/0702411.

[12]  Jian Ding,et al.  Total variation cutoff in birth-and-death chains , 2008, 0801.2625.

[13]  Guan-Yu Chen,et al.  The cutoff phenomenon for ergodic Markov processes , 2008 .

[14]  R. Fernández,et al.  Abrupt Convergence and Escape Behavior for Birth and Death Chains , 2009, 0903.1832.

[15]  Guan-Yu Chen,et al.  Spectral computations for birth and death chains , 2013 .

[16]  Guan-Yu Chen,et al.  Comparison of Cutoffs Between Lazy Walks and Markovian Semigroups , 2013, Journal of Applied Probability.