Edgeworth expansions of suitably normalized sample mean statistics for atomic Markov chains

This paper is devoted to the problem of estimating functionals of type μ(f) = ∫ f d μ from observations drawn from a positive recurrent atomic Markov chain X = (X_n)_n∈ℕ with stationary distribution μ. The properties of different estimators are studied. Beyond an accurate estimation of their bias, the estimation of their asymptotic variance is considered. We also show that the results of Malinovskii (1987) on the validity of the formal Edgeworth expansion for sample mean statistics of type T_n = n^-1 ∑_i=1ⁿ f (X_i) extend to their studentized versions, normalized by the asymptotic variance estimates we consider.