Nonparametric estimation of the stationary density and the transition density of a Markov chain

In this paper, we study first the problem of nonparametric estimation of the stationary density f of a discrete-time Markov chain (X-i). We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables us to estimate the density g of (X-i, Xi+1) and so to provide an adaptive estimator of the transition density pi = g/f. We give bounds in L-2 norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. Some examples and simulations are also provided. (c) 2007 Published by Elsevier B.V.