On the central limit theorem for some birth and death processes

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.