Geodesic
On the central limit theorem for some birth and death processes
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 65 (2011) no. 1.

Voir la notice de l'article provenant de la source Library of Science

Suppose that {Xn: n ≥ 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∑_n=0^N V (X_n) converge in law to a normal random variable, as N →+∞. 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.
Keywords: Central limit theorem, Markov chain, Lamperti’s problem, birth and death processes, Kipnis-Varadhan theory, spectral gap 
@article{AUM_2011_65_1_a5,
     author = {Chojecki, Tymoteusz},
     title = {On the central limit theorem for some birth and death processes},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     publisher = {mathdoc},
     volume = {65},
     number = {1},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2011_65_1_a5/}
}
TY  - JOUR
AU  - Chojecki, Tymoteusz
TI  - On the central limit theorem for some birth and death processes
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2011
VL  - 65
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2011_65_1_a5/
LA  - en
ID  - AUM_2011_65_1_a5
ER  - 
%0 Journal Article
%A Chojecki, Tymoteusz
%T On the central limit theorem for some birth and death processes
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2011
%V 65
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2011_65_1_a5/
%G en
%F AUM_2011_65_1_a5
Chojecki, Tymoteusz. On the central limit theorem for some birth and death processes. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 65 (2011) no. 1. http://geodesic.mathdoc.fr/item/AUM_2011_65_1_a5/

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

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

[3] 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).

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

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

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

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

[8] 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.

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

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

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