Bernoulli sums and Rényi entropy inequalities
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We investigate the Rényi entropy of independent sums of integer valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the Rényi entropy, for Poisson-Bernoulli variables. As applications we prove that a discrete “min-entropy power” is super additive on independent variables up to a universal constant, and give new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the “Poisson regime”.
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