On the central limit theorem for some birth and death processes

Tymoteusz Chojecki


Suppose that \(\{Xn: n \geq 0\}\) is a stationary Markov chain and \(V\) is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if \(Y_n :=
N^{-1/2}\sum_{n=0}^N V (X_n)\) converge in law to a normal random variable, as \(N \to+\infty\). For a stationary Markov chain with the \(L^2\) spectral gap the theorem holds for all \(V\) such that \(V (X_0)\) is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables \(V\) for which the CLT holds for a class of birth and death chains whose dynamics has no spectral gap, so that Gordin’s result cannot be used and the result follows from an application of Kipnis-Varadhan theory.


Central limit theorem; Markov chain; Lamperti’s problem; birth and death processes; Kipnis-Varadhan theory; spectral gap

Full Text:



Chen, M., Eigenvalues, Inequalities, and Ergodic Theory, Springer-Verlag, London, 2005.

Chung, K. L., Markov Chains with Stationary Transition Probabilities, 2nd edition, Springer-Verlag, Berlin, 1967.

De Masi, A., Ferrari, P. A., Goldstein, S. and Wick, W. D., An invariance principle for reversible Markov processes. Applications to random motions in random environments, J. Statist. Phys. 55 (1989).

Doeblin, W., Sur deux proble‘mes de M. Kolmogoroff concernant les chaines d´enombrables, Bull. Soc. Math. France 66 (1938), 210-220.

Durrett, R., Probability Theory and Examples, Wadsworth Publishing Company, Belmont, 1996.

Feller, W., An Introduction to Probability Theory and its Applications, Vol. II. Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.

Gordin, M. I., The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739-741 (Russian).

Kipnis, C., Varadhan, S. R. S., Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. Math. Phys. 104, no. 1 (1986), 1-19.

Liggett, T., Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Grund. der Math. Wissen., 324, Springer-Verlag, Berlin, 1999.

Menshikov, M.,Wade, A.,Rate of escape and central limit theorem for the supercritical Lamperti problem, Stochastic Process. Appl. 120 (2010), 2078-2099.

Olla, S., Notes on Central Limits Theorems for Tagged Particles and Diffusions in Random Environment, Etats de la recherche: Milieux Aleatoires CIRM, Luminy, 2000.

DOI: http://dx.doi.org/10.17951/a.2011.65.1.21-31
Data publikacji: 2016-07-25 18:17:30
Data złożenia artykułu: 2016-07-25 15:18:02


Total abstract view - 216
Downloads (from 2020-06-17) - PDF - 33



  • There are currently no refbacks.

Copyright (c) 2011 Tymoteusz Chojecki