• Corpus ID: 226976761

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… 

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References

SHOWING 1-10 OF 64 REFERENCES

The Extreme Points of Subsets of s-Concave Probabilities and a Geometric Localization Theorem

It is proved that the extreme points of the set of s-concave probability measures satisfying a linear constraint are some Dirac measures and some s-affine probabilities supported by a segment, which gives a new approach to a localization theorem.

Concentration inequalities for $s$-concave measures of dilations of Borel sets and applications

We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in the Euclidean space by a $s$-concave probability measure. Our result gives a common generalization of an

Poisson processes and a log-concave Bernstein theorem

We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in

Negative dependence and the geometry of polynomials

We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers

Thin Shell Implies Spectral Gap Up to Polylog via a Stochastic Localization Scheme

We consider the isoperimetric inequality on the class of high-dimensional isotropic convex bodies. We establish quantitative connections between two well-known open problems related to this

A Brunn-Minkowski inequality for the integer lattice

A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the

Log-Concavity and Strong Log-Concavity: a review.

A new proof of Efron's theorem is provided using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013) and along the way connections between log-concavity and other areas of mathematics and statistics are reviewed.

From Steiner Formulas for Cones to Concentration of Intrinsic Volumes

A systematic technique for studying conic intrinsic volumes using methods from probability, based on a general Steiner formula for cones, which leads to new identities and bounds for the intrinsic volumes of a cone, including a near-optimal concentration inequality.

Khinchin-type inequalities via Hadamard's factorisation

We prove Khinchin-type inequalities with sharp constants for type L random variables and all even moments. Our main tool is Hadamard’s factorisation theorem from complex analysis, combined with
...