Efficiency of Empirical Estimators for Markov Chains

Suppose we observe a uniformly ergodic Markov chain with unknown transition distribution. The empirical estimator for a linear functional of the (invariant) joint distribution of two successive observations is defined using the pairs of successive observations. Its efficiency is proved using a martingale approximation. As a corollary we show efficiency of the empirical joint distribution function in the sense of a functional convolution theorem.