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2011 | 65 | 1 |
Tytuł artykułu

On the central limit theorem for some birth and death processes

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EN
Abstrakty
EN
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.
Rocznik
Tom
65
Numer
1
Opis fizyczny
Daty
wydano
2011
online
2016-07-25
Twórcy
Bibliografia
  • 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.
Typ dokumentu
Bibliografia
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bwmeta1.element.ojs-doi-10_17951_a_2011_65_1_21-31
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