Abstract

Logarithmic Sobolev and transportation cost inequalities have proved to be relevant tools in the study of concentration inequalities for various models of statistical interests. Stein’s method is a powerful technique for proving central limit theorems. In this talk, we develop new connections between Stein’s approximation method and logarithmic Sobolev and transport inequalities by introducing a class of functional inequalities involving the relative entropy, the Stein (factor or) matrix, the relative Fisher information and the Wasserstein distance. For the Gaussian model, these results improve upon the classical logarithmic Sobolev inequality and the Talagrand quadratic transportation cost inequality. The new inequalities produce bounds for normal entropic convergence expressed in terms of the Stein discrepancy applicable to various examples of multidimensional functionals. Joint work with Ivan Nourdin and Giovanni Peccati.