On the central limit theorem for some birth and death processes

Tymoteusz Chojecki

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011), / Harvested from The Polish Digital Mathematics Library

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Résumé

Suppose that ${X n : n \geq 0}$ 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.

Publié le : 2011-01-01
EUDML-ID : urn:eudml:doc:289828

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Tymoteusz Chojecki. On the central limit theorem for some birth and death processes. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2011_65_1_21-31/

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