Prediction in invertible linear processes

We construct root-nn consistent plug-in estimators for conditional expectations of the form E(h(Xn+1,…,Xn+m)|X1,…,Xn)E(h(Xn+1,…,Xn+m)|X1,…,Xn) in invertible linear processes. More specifically, we prove a Bahadur-type representation for such estimators, uniformly over certain classes of not necessarily bounded functions hh. We obtain in particular a uniformly root-nn consistent estimator for the mm-dimensional conditional distribution function. The proof uses empirical process techniques.