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We extend the dimension free Talagrand inequalities for convex distance\cite{talagrand:1995} using an extension of Marton's weak transport\cite{marton:1996a} to other metrics than the Hamming distance. We study thedual form of these weak transport inequalities for the euclidian norm and provethat it implies sub-gaussianity and convex Poincar\'e inequality\cite{bobkov:gotze:1999a}. We obtain new weak transport inequalities for nonproducts measures extending the results of Samson in \cite{samson:2000}. Manyexamples are provided to show that the euclidian norm is an appropriate metricfor classical time series. Our approach, based on trajectories coupling, ismore efficient to obtain dimension free concentration than existing contractiveassumptions \cite{djellout:guillin:wu:2004,marton:2004}. Expressing theconcentration properties of the ordinary least square estimator as aconditional weak transport problem, we derive new oracle inequalities with fastrates of convergence in dependent settings.

Weak transport inequalities and applications to exponential inequalities and oracle inequalities