Weighted Poincaré-type inequalities for Cauchy and other convex measures

Abstract

Brascamp--Lieb-type, weighted Poincaré-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.

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)

References in corpus (1)

• Asymptotics of the fast diffusion equation via entropy estimates

Cited by in corpus (4)

• Comparison of Markov chains via weak Poincaré inequalities with application to pseudo-marginal MCMC
• Concentration of empirical distribution functions with applications to non-i.i.d. models
• Gradient flows of $(K,N)$-convex functions with negative $N$
• Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities