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 (Xi). 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 (Xi,Xi+1) and so to provide an adaptive estimator of the transition density [pi]=g/f. We give bounds in L2 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.