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.
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Olivier Wintenberger. Weak transport inequalities and applications to exponential and oracle inequalities. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, 20, pp.114. ⟨10.1214/EJP.v20-3558⟩. ⟨hal-01263366⟩

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