# Geometric and functional inequalities for log-concave probability sequences

@article{Marsiglietti2020GeometricAF, title={Geometric and functional inequalities for log-concave probability sequences}, author={Arnaud Marsiglietti and James Melbourne}, journal={arXiv: Probability}, year={2020} }

We investigate various geometric and functional inequalities for the class of log-concave probability sequences. We prove dilation inequalities for log-concave probability measures on the integers. A functional analog of this geometric inequality is derived, giving large and small deviation inequalities from a median, in terms of a modulus of regularity. Our methods are of independent interest, we find that log-affine sequences are the extreme points of the set of log-concave sequences…

## 8 Citations

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. We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an…

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A discrete version of the notion of degree of freedom is utilized to prove a sharp min-entropy-variance inequality for integer valued log-concave random variables and it is shown that the geometric distribution minimizes the min-Entropy within the class of log- Concave probability sequences with variance.

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We establish a reversal of Lyapunov’s inequality for monotone log-concave sequences, settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A strengthened version of the same conjecture is…

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A remarkable conjecture of Feige (2006) asserts that for any collection of n independent non-negative random variables X 1 , X 2 , . . . , X n , each with expectation at most 1, P ( X < E [ X ] + 1)…

### Moments, Concentration, and Entropy of Log-Concave Distributions

- EconomicsArXiv
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We utilize and extend a simple and classical mechanism, combining log-concavity and majorization in the convex order to derive moments, concentration, and entropy inequalities for log-concave random…

### A discrete complement of Lyapunov's inequality and its information theoretic consequences

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A reversal of Lyapunov’s inequality for monotone log-concave sequences is established, settling a conjecture of Havrilla-Tkocz and Melbourne-T kocz, and several information theoretic inequalities as consequences are derived.

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We give an alternative proof for discrete Brunn-Minkowski type inequalities, recently obtained by Halikias, Klartag and the author. This proof also implies stronger weighted versions of these…

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