Weighted Poincar\'{e}-type inequalities for Cauchy and other convex measures

Sergey G. Bobkov; Michel Ledoux

doi:10.1214/08-AOP407

Weighted Poincar\'{e}-type inequalities for Cauchy and other convex measures

Probability 2009-06-10 v1

Authors: Sergey G. Bobkov , Michel Ledoux

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Abstract

Brascamp--Lieb-type, weighted Poincar\'{e}-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general $\kappa$ -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.

Keywords

weighted Poincaré inequality Cauchy distribution convex measure κ-concave measure measure concentration large deviation Cheeger isoperimetric inequality

Cite

@article{arxiv.0906.1651,
  title  = {Weighted Poincar\'{e}-type inequalities for Cauchy and other convex measures},
  author = {Sergey G. Bobkov and Michel Ledoux},
  journal= {arXiv preprint arXiv:0906.1651},
  year   = {2009}
}

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Published in at http://dx.doi.org/10.1214/08-AOP407 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Weighted Poincar\'{e}-type inequalities for Cauchy and other convex measures

Abstract

Keywords

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