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Register a new userEntropic CLT for smoothed convolutions and associated entropy bounds
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We explore an asymptotic behavior of entropies for sums of independent random variables that are convolved with a small continuous noise.
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We explore an asymptotic behavior of densities of sums of independent random variables that are convoluted with a small continuous noise.
In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced by Bigot, Cazelles and Papadakis (2019) as a regularization of Wasserstein barycenters first presented by Agueh and Carlier (2011). After characterizing these barycenters in terms of a system of Monge-Amp`ere equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters.
Two-sided bounds are explored for concentration functions and Renyi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities.
We investigate the role of convexity in Renyi entropy power inequalities. After proving that a general Renyi entropy power inequality in the style of Bobkov-Chistyakov (2015) fails when the Renyi parameter $rin(0,1)$, we show that random vectors with $s$-concave densities do satisfy such a Renyi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for Renyi entropies of order $rin(0,1)$ for log-concave densities and for compactly supported, spherically symmetric and unimodal densities, complementing a celebrated result of Barron (1986). Additionally, we give an entropic characterization of the class of $s$-concave densities, which extends a classical result of Cover and Zhang (1994).
Let $(X_1 , ldots , X_d)$ be random variables taking nonnegative integer values and let $f(z_1, ldots , z_d)$ be the probability generating function. Suppose that $f$ is real stable; equivalently, suppose that the polarization of this probability distribution is strong Rayleigh. In specific examples, such as occupation counts of disjoint sets by a determinantal point process, it is known~cite{soshnikov02} that the joint distribution must approach a multivariate Gaussian distribution. We show that this conclusion follows already from stability of $f$.
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