On the central limit theorem for some birth and death processes

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 converge in law to a normal random variable, as N → +∞. For a stationary Markov chain with the L2 spectral gap the theorem holds for all V such that V (X0) 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.