Abstract
We extend the dimension free Talagrand inequalities for convex distance using an extension of Marton’s weak transport to other metrics than the Hamming distance. We study the dual form of these weak transport inequalities for the euclidian norm and prove that it implies sub-gaussianity and convex Poincaré inequality. We obtain new weak transport inequalities for non products measures extending the results of Samson. Many examples are provided to show that the euclidian norm is an appropriate metric for classical time series. Our approach, based on trajectories coupling, is more efficient to obtain dimension free concentration than existing contractive assumptions. Expressing the concentration properties of the ordinary least square estimator as a conditional weak transport problem, we derive new oracle inequalities with fast rates of convergence in dependent settings.
Highlights
In his remarkable paper [49], Talagrand proved that convex distances have dimensionfree concentration properties
We introduce the notion of Γd,d (p)-weak dependence in Definition 3.14 to assert the existence of a coupling between the trajectories (Xi+1, . . . , Xn), given the same past Xk, k < i, controlling possible deviations in the present Xi through an auxiliary metric d satisfying d ≤ M d, M > 0
The weak transport approach provides dimension-free concentration properties of ARMA processes under minimal assumptions. It is sufficient for extending fast rates of convergence in statistical applications from the classical iid setting to the Γ(2)-weak dependence one
Summary
In his remarkable paper [49], Talagrand proved that convex distances have dimensionfree concentration properties. Samson’s results yield fast convergence rates of order n−1 in statistical applications for uniformly mixing sequences; see [2] His approach relies on the maximal coupling properties and cannot be extended in a direct way to more general dependent settings because the maximal coupling exists only for Hamming’s distance; see [19]. In the case p = 2, we prove the first dimension-free concentration result for ARMA processes under the minimal dependence assumption that the stationary distribution exists. The weak transport approach provides dimension-free concentration properties of ARMA processes under minimal assumptions. It is sufficient for extending fast rates of convergence in statistical applications from the classical iid setting to the Γ(2)-weak dependence one.
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