Weighted poincaré-type inequalities for cauchy and other convex measures

By Sergey G. Bobkov, Michel Ledoux. 2009

127 citations recorded by the source corpus.

Brascamp-Lieb-type, weighted Poincare-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8c760bb9caf48b2b403c5fdca79707024380e12a

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Log-Concavity and Strong Log-Concavity: a review (2014, background): “The first inequality in (d) is referred in Bobkov and Ledoux (2009) as a “weighted Poincaré-type inequality” for convex (or s-concave with negative parameter s) measures.”
(K, N)-Convexity and the Curvature-Dimension Condition for Negative N (2013, background): “aces, such as Borell’s convex (or 1/N-concave) measures (see [Bo], [BrLi], and the paragraph following Theorem 4.8) and a connection with Barenblatt solutions to the porous medium equation (see [Ot], [BoLe]). Acknowledgements. I am grateful to Kazumasa Kuwada for valuable suggestions and discussions”
Spectral gap for spherically symmetric log-concave probability measures, and beyond (2014, methodology): “Motivated by an open question raised in [6], our second choice of radial weight will be σ(2)(x) := 1 1 + ‖x‖2 , which satisfies also Assumption A.”
Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, background): “…measures μ on ( R n , |·| ) satisfying the CD(0 , N ) condition with 1 N ∈ [ −∞ , 1 /n ] have already been studied by Borell [12] (cf. [14]), who dubbed them convex (or 1 N -concave) measures; see [9, 11, 31, 49, 53] for a more detailed account of these measures and their useful properties.”
Coercive Inequalities on Metric Measure Spaces (2009, background): “[11]) which in particular allows us to extend recent results of [9] where convexity ideas in Euclidean spaces were used.”
Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry (2010, methodology): “Our first example corresponds to the convex case discussed by Bobkov and Ledoux [23].”