Speed of stability for birth-death processes

doi:10.1007/s11464-010-0068-7

Front. Math. China

2010, Vol. 5

Issue (3) : 379-515 https://doi.org/10.1007/s11464-010-0068-7

Research articles

Speed of stability for birth-death processes

Mu-Fa CHEN,

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, China;

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Abstract This paper is a continuation of the study on the stability speed for Markov processes. It extends the previous study of the ergodic convergence speed to the non-ergodic one, in which the processes are even allowed to be explosive or to have general killings. At the beginning stage, this paper is concentrated on the birth-death processes. According to the classification of the boundaries, there are four cases plus one more having general killings. In each case, some dual variational formulas for the convergence rate are presented, from which, the criterion for the positivity of the rate and an approximating procedure of estimating the rate are deduced. As the first step of the approximation, the ratio of the resulting bounds is usually no more than 2. The criteria as well as basic estimates for more general types of stability are also presented. Even though the paper contributes mainly to the non-ergodic case, there are some improvements in the ergodic one. To illustrate the power of the results, a large number of examples are included.

Keywords Birth-death process speed of stability first eigenvalue variational formula criterion and basic estimates approximating procedure duality killing

Issue Date: 05 September 2010

Cite this article:

Mu-Fa CHEN. Speed of stability for birth-death processes[J]. Front. Math. China, 2010, 5(3): 379-515.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-010-0068-7
https://academic.hep.com.cn/fmc/EN/Y2010/V5/I3/379

	Chen A, Pollett P, Zhang H, Cairns B. Uniquenesscriteria for continuous-time Markov chains with general transitionstructure. Adv Appl Prob, 2005, 37(4): 1056―1074 doi: 10.1239/aap/1134587753
	Chen M F. Exponential L²-convergence and L²-spectral gap for Markov processes. Acta Math Sin (New Ser), 1991, 7(1): 19―37 doi: 10.1007/BF02582989
	Chen M F. Estimation of spectral gap for Markov chains. Acta Math Sin (New Ser), 1996, 12(4): 337―360 doi: 10.1007/BF02106789
	Chen M F. Analytic proof of dual variational formula for the first eigenvaluein dimension one. Sci in China, A, 1999, 42(8): 805―815
	Chen M F. The principal eigenvalue for jump processes. Acta Math Sin (Eng Ser), 2000, 16(3): 361―368
	Chen M F. Explicit bounds of the first eigenvalue. Sci Chin, Ser A, 2000, 43(10): 1051―1059
	Chen M F. Variational formulas and approximation theorems for the first eigenvaluein dimension one. Sci Chin, Ser A, 2001, 44(4): 409―418
	[Collectionof papers, 1993—]math.bnu.edu.cn/?chenmf/maineng.htm
	Chen M F. Variational formulas of Poincaré-type inequalities for birth-deathprocesses. Acta Math Sin (Eng Ser), 2003, 19(4): 625―644
	Chen M F. From Markov Chains to Non-equilibrium Particle Systems. 2nd ed. Singapore: World Scientific, 2004 doi: 10.1142/9789812562456
	Chen M F. Capacitary criteria for Poincaré-type inequalities. Potential Theory, 2005, 23(4): 303―322
	Chen M F. Eigenvalues, Inequalities, and Ergodic Theory. London: Springer, 2005
	Chen M F, Wang F Y. Estimation of spectral gapfor elliptic operators. Trans Amer MathSoc, 1997, 349(3): 1239―1267
	Chen M F, Wang F Y. Cheeger’s inequalitiesfor general symmetric forms and existence criteria for spectral gap. (Abstract) Chin Sci Bull, 1998, 43(18): 1516―1519; Ann Prob, 2000, 28(1): 235―257
	Chen M F, Zhang Y H, Zhao X L. Dual variational formulas for the first Dirichlet eigenvalueon half-line. Sci China, Ser A, 2003, 46(6): 847―861
	Cox J T, R?sler U. A duality relation for entranceand exit laws for Markov processes. StochProc Appl, 1983, 16: 141―156
	Dobrushin R L. On conditions of regularity of stationary Markov processes with adenumerable number of possible states. Uspehi Matem Nauk (NS), 1952, 7(6): 185―191 (in Russian)
	Fukushima M, Uemura T. Capacitary bounds of measuresand ultracontractivity of time changed processes. J Math Pure et Appliquees, 2003, 82(5): 553―572
	Hou Z T, Liu Z M, Zhang H J, Li J P, Zhou J Z, Yuan C G. Birth-death Processes. Changsha: Hunan Sci and Tech Press, 2000 (in Chinese)
	Hou Z T, Zhou J Z, Zhang H J, Liu Z M, Xiao G N, Chen A Y, Fei Z L. The Q-matrix Problem for Markov Chains. Changsha: Hunan Sci and Tech Press, 1994 (in Chinese)
	Jin H Y. Estimating the constant in L^p-Poincaré inequality. Master Thesis. Beijing: Beijing Normal Univ, 2006 (in Chinese)
	Karlin S, McGregor J L. The differential equationsof birth-and-death processes, and the Stieltjes moment problem. Trans Amer Math Soc, 1957, 85: 589―646
	Karlin S, McGregor J. The classification of birthand death processes. Trans Amer Math Soc, 1957, 86(2): 366―400
	Kijima M. MarkovProcesses for Stochastic Modeling. London: Chapman & Hall, 1997
	Mao Y H. Nash inequalities for Markov processes in dimension one. Acta Math Sin (Eng Ser), 2002, 18(1): 147―156
	Mao Y H, Xia L Y. Spectral gap for jump processesby decomposition method. Front Math China, 2009, 4(2): 335―347
	Maz’ya V G. Sobolev Spaces. Berlin: Springer, 1985
	Miclo L. Anexample of application of discrete Hardy’s inequalities. Markov Processes Relat Fields, 1999, 5: 319―330
	Muckenhoupt B. Hardy’sinequality with weights. Studia Math, 1972, XLIV: 31―38
	Opic B, Kufner A. Hardy-type Inequalities. New York: Longman, 1990
	Shao J H, Mao Y H. Estimation of the Dirichleteigenvalue of birth-death process on trees. Acta Math Sin (Chinese Ser), 2007, 50(3): 507―516 (in Chinese)
	Shiozawa Y, Takeda M. Variational formula for Dirichletforms and estimates of principal eigenvalues for symmetric α-stableprocesses. Potential Analysis, 2005, 23: 135―151
	Sirl D, Zhang H, Pollett P. Computable bounds for the decay parameter of a birthdeathprocess. J Appl Prob, 2007, 44(2): 476―491 doi: 10.1239/jap/1183667415
	van Doorn E A. Stochastic Monotonicity and Queuing Applications of Birth-Death Processes. Lecture Notes in Statistics, Vol 4. Berlin: Springer-Verlag, 1981
	van Doorn E A. Conditions for exponential ergodicity and bounds for the decay parameterof a birth-death process. Adv Appl Prob, 1985, 17: 514―530 doi: 10.2307/1427118
	van Doorn E A. Representations and bounds for zeros of orthogonal polynomials andeigenvalues of sign-symmetric tri-diagonal matrices. J Approx Th, 1987, 51: 254―266 doi: 10.1016/0021-9045(87)90038-4
	van Doorn E A. Representations for the rate of convergence of birth-death processes. Theory Probab Math Statist, 2002, 65: 37―43
	Wang J. FirstDirichlet eigenvalue of transient birth-death processes. 2008, preprint
	Wang J. Poincaré-typeinequalities for transient birth-death processes. 2008, preprint
	Wang J. Functionalinequalities for transient birth-death processes and their applications. 2008, preprint
	Wang Z K. The ergodicity and zero-one law for birth and death processes. Acta Sci Nankai Univ, 1964, 5(5): 93―102 (in Chinese)
	Wang Z K, Yang X Q. Birth and Death Processesand Markov Chains. Berlin: Springer and Beijing: Sci Press, 1992
	Zeifman A I. Some estimates of the rate of convergence for birth and death processes. J Appl Prob, 1991, 28: 268―277 doi: 10.2307/3214865
	Zhang X. TheEstimation for the First Eigenvalue of Schr?dinger Operatorsand a Class of Geometric Inequalities. Ph D Thesis. Beijing: Beijing Normal University, 2007 (in Chinese)

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