Weighted Poincaré-type inequalities for Cauchy and other convex measures

Details Weighted Poincaré-type inequalities for Cauchy and other convex measures Weighted Poincaré-type inequalities for Cauchy and other convex measures

2009-06-09

arxiv.org/abs/0906.1651v1

Annals of Probability 2009, Vol. 37, No. 2, 403-427

Mathematics

Probability

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

Scientific paper

10.1214/08-AOP407

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.

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