Efficient estimation of the stationary distribution for exponentially ergodic Markov chains

Abstract

In a classical paper by Dvoretsky, Kiefer and Wolfowitz the asymptotic minimaxity of the empirical distribution function in case of i.i.d. observations X1,X2,…,Xn of the random variable X has been shown. If X1,X2,…,Xn,… is only a stationary sequence we still could use the empirical distribution function as an estimator of the (continuous) stationary distribution F. But the question of its asymptotic efficiency arises in this case. Under some additional assumptions (stationary homogeneous exponentially ergodic Markov sequence) we show that the empirical distribution function is an efficient estimator in a local asymptotic minimax sense.Using the bounded subconvex loss function g(supt√n|F̂n(t) − F(t)|) with g-bounded, increasing, the local asymptotic minimax bound equals Eg(sup |Y(t)|) where Y(t) is a certain Gaussian process.