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Quantitative limit theorems via relative log-concavity
Arturo Jaramillo, James Melbourne
In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures and $ν…
Quantitative form of Ball's Cube slicing in and equality cases in the min-entropy power inequality
James Melbourne, Cyril Roberto
We prove a quantitative form of the celebrated Ball's theorem on cube slicing in and obtain, as a consequence, equality cases in the min-entropy power inequality. In…
Concentration Inequalities for Ultra Log-Concave Distributions
Heshan Aravinda, Arnaud Marsiglietti, James Melbourne
We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration b…
Concentration functions and entropy bounds for discrete log-concave distributions
Sergey G. Bobkov, Arnaud Marsiglietti, James Melbourne
Two-sided bounds are explored for concentration functions and Rényi entropies in the class of discrete log-concave probability distributions. They are used to derive certain varian…
Further investigations of Rényi entropy power inequalities and an entropic characterization of s-concave densities
Jiange Li, Arnaud Marsiglietti, James Melbourne
We investigate the role of convexity in Rényi entropy power inequalities. After proving that a general Rényi entropy power inequality in the style of Bobkov-Chistyakov (2015) fails…
Rearrangement and Prekopa-Leindler type inequalities
James Melbourne
We investigate the interactions of functional rearrangements with Prekopa-Leindler type inequalities. It is shown that that a general class of integral inequalities tighten on rear…