On the central limit theorem for some birth and death processes

Suppose that $\{Xn:n\ge 0\}$ 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 $N\to +\mathrm{\infty }$. 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.