Recursive estimation of the transition distribution function of a Markov process: A symptotic normality

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 recursive nonparametric estimate of the one-step transition distribution function is shown to be asymptotically normal. 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.