On the CLT for stationary Markov chains with trivial tail sigma field (Q2686009)

Let \((\xi_i)\) be a strictly stationary Markov chain with invariant measure \(\pi\), and \(f\) a measurable function such that \(\int f^2 d\pi<\infty\) and \(\int f d\pi=0\). The main objects of study in the present paper are the partial sums \(S_n=\sum_{i=1}^nf(\xi_i)\). The author shows that if the two-sided tail sigma field of \((\xi_i)\) is trivial and \(\sup_n E(S_n^2)/n<\infty\), then \(S_n/\sqrt{n}\) satisfies a central limit theorem. The class of stationary Markov chains that satisfy the condition of having a trivial two-sided tail sigma field includes absolutely regular Markov chains and interlaced mixing Markov chains, both of which are discussed in detail by the author. The central limit theorem presented here avoids the need for any detailed computation of convergence rates of mixing coefficients, and any random centering.