Homogeneous birth and death processes with a finite number of states are studied. We analyze the slowest and fastest rates of convergence to the limit mode. Estimates of these bounds for some classes of mean-field models are obtained. The asymptotics of the convergence rate for some models of chemical kinetics is studied in the case where the number of system states tends to infinity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Diaconis and L. Salloff-Coste, “Walks on generating sets of Abelian groups,” Probab. Theory Relat. Fields, 105, 393–421 (1996).
E.A. Van Doorn, “Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process,” Adv. Appl. Probab., 17, 514–530 (1985).
E. A. Doorn and A. I. Zeifman, “Extinction probability in a birth-death process with killing,” J. Appl. Probab., 42, 185–198 (2005).
R. Fernández, J. Frőhlich, and A.D. Sokal, Random Walks, Critical Phenomenon and Triviality in Quantum Field Theory, Springer, Berlin (1992).
B. Granovsky and A. I. Zeifman, “The decay function of nonhomogeneous birth-death processes, with application to mean-field models,” Stoch. Process. Appl., 72, 105–120 (1997).
B. Granovsky and A. I. Zeifman, “The N-limit of spectral gap of a class of birth-death Markov chains,” Appl. Stoch. Models Bus. Ind., 16, 235–248 (2000).
B. Granovsky and A. I. Zeifman, “On the lower bound of the spectrum of some mean-field models,” Theory Prob. Appl., 49, 148–155 (2005).
T. M. Liggett, Interacting Particle Systems, Springer, New York (2005).
A.Yu. Mitrophanov, “Note on Zeifman’s bounds on the rate of convergence for birth-death processes,” J. Appl. Probab., 41, 593–596 (2004).
P.K. Polett and A. Vassalo, “Diffusion approximations for some simple chemical reaction schemes,” Adv. Appl. Probab., 24, 875–893 (1992).
A. I. Zeifman, “Some estimates of the rate of convergence for birth and death processes,” J. Appl. Probab., 28, 268–277 (1991).
A. I. Zeifman, “Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes,” Stoch. Proc. Appl., 59, 157–173 (1995).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, Vol. 20, pp. 140–148, 2007
Rights and permissions
About this article
Cite this article
Zeifman, A.I., Panfilova, T.L. On Convergence Rate Estimates for Some Birth and Death Processes. J Math Sci 221, 616–622 (2017). https://doi.org/10.1007/s10958-017-3254-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-017-3254-2