We prove that if (P-x)(x is an element of X) is a family of probability measures which satisfy the log-Sobolev inequality and whose pairwise chi-squared divergences are uniformly bounded, and mu is any mixing distribution on X, then the mixture integral P(x)d mu(x) satisfies a log-Sobolev inequality. In various settings of interest, the resulting log-Sobolev constant is dimension-free. In particular, our result implies a conjecture of Zimmermann and Bardet et al. that Gaussian convolutions of measures with bounded support enjoy dimension-free log-Sobolev inequalities. (C) 2021 Elsevier Inc. All rights reserved.
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