Iterated Barron density estimators

Barron estimators are modified histograms depending on some reference proba bility density. They have been proved to have good consistency properties a ccording to several information theoretical criteria. In this Note, Barren estimators are viewed as dynamical systems in a functional space with the r eference density as initial state. Properties of stationary densities of th ese systems are investigated. It is then proved, under some assumptions, th at the trajectories become almost surely stationary after a finite number o f iterations. (C) 2001 Academie des sciences/Editions scientifiques et medi cales Elsevier SAS.