On the central limit theorem for some birth and death processes

Tymoteusz Chojecki

Abstract


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.

Keywords


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

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References


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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


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