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Free Stein kernels and an improvement of the free logarithmic Sobolev inequality (1605.05828v1)
Published 19 May 2016 in math.OA and math.PR
Abstract: We introduce a free version of the Stein kernel, relative to a semicircular law. We use it to obtain a free counterpart of the HSI inequality of Ledoux, Peccatti and Nourdin, which is an improvement of the free logarithmic Sobolev inequality of Biane and Speicher, as well as a rate of convergence in the (multivariate) entropic free Central Limit Theorem. We also compute the free Stein kernels for several relevant families of self-adjoint operators.