Estimation of Transition Distribution Function and its Quantiles in Markov Processes: Strong Consistency and Asymptotic Normality

Abstract

Let X1,…,Xn+1 be the first n+1 random variables from a strictly stationary Markov process which satisfies certain additional regularity conditions. On the basis of these random variables, a nonparametric estimate of the one-step transition distribution function is shown to be uniformly (in the main argument) strongly consistent and asymptotically normal. Furthermore, for any p∈(0, 1), a natural nonparametric estimate of the p-th quantile of the distribution function just mentioned is proven to be strongly consistent. The class of Markov processes studied includes the Markov processes usually considered in the literature; namely, processes which either satisfy Doeblin’s hypothesis, or, more generally, are geometrically ergodic.