  Suppose that X_n, n>=0 is a stationary Markov chain and V is a certain
function on a phase space of the chain, called an observanle. We say that the
observable satisfies the central limit theorem (C.L.T.) if
Y_n:=N^{-1/2}\sum_{n=0}^NV(X_n) converge in law to a normal random variable, as
N goes to infinity. 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. The purpose of this article is to characterize a family of observables
V for which the C.L.T. 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.
