Weak transport inequalities and applications to exponential and oracle inequalities

Abstract

We study the dimension-free inequalities, see Talagrand [49], for non-product measures extending Marton’s [39] weak transport from the Hamming distance to other metrics. The Euclidian norm is proved to be appropriate for dealing with non-product measures associated with classical time series. Our approach to address dependence, based on coupling of trajectories, weakens previous contractive arguments used in [20] and [41]. Following Bobkov-Götze’s [10] approach, we derive sub-Gaussianity and a convex Poincaré inequality for non-product measures that are not uniformly mixing, extending the Samson’s [48] results. Such dimension-free inequalities are useful for applications in statistics. Expressing the concentration properties of the ordinary least squares estimator as a weak transport problem, we obtain new oracle inequalities with fast rates of convergence for classical time series models.