On the rate of convergence in the invariance principle for real-valued functions of Doeblin processes

The speed of convergence in the functional central limit theorem (or invariance principle) for partial sum processes based on real-valued functions of Markov processes satisfying Doeblin's condition is studied where Prokhorov's metric is used to measure the distance between probability distributions on C([0, 1]). For underlying variables with finite absolute moments of an order greater than two and less than five the rate obtained is the same as that in the case of independent and identically distributed random variables which is known to be exact. The proof is based on Gordin's decomposition method and the martingale version of Skorokhod's embedding. A non-uniform Berry-Esséen estimate for the maximum absolute value of partial sums of bounded functions is also established.