Abstract
We investigate the Rényi entropy of sums of independent integer-valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the Rényi entropy for sums of independent Bernoulli random variables. As applications, we prove that a discrete “min-entropy power” is superadditive with respect to convolution modulo a universal constant, and give new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the “Poisson regime”.
Funding Statement
The last author was supported by the Labex MME-DII funded by ANR, reference ANR-11-LBX-0023-01 and ANR-15-CE40-0020-03 - LSD - Large Stochastic Dynamics, and the grant of the Simone and Cino Del Duca Foundation, France.
Acknowledgements
The authors thank Arnaud Marsiglietti for stimulating discussion and in particular suggesting the connection to the Littlewood-Offord question of [22], as well as an anonymous reviewer whose careful reading and suggestions have improved this article, and to whom Proposition 2.4 is to be credited.
Citation
Mokshay Madiman. James Melbourne. Cyril Roberto. "Bernoulli sums and Rényi entropy inequalities." Bernoulli 29 (2) 1578 - 1599, May 2023. https://doi.org/10.3150/22-BEJ1511
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