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Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities
Abstract We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a
Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures
We discuss an approach, based on the Brunn–Minkowski inequality, to isoperimetric and analytic inequalities for probability measures on Euclidean space with logarithmically concave densities. In
Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution
Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every
From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities
Abstract. We develop several applications of the Brunn—Minkowski inequality in the Prékopa—Leindler form. In particular, we show that an argument of B. Maurey may be adapted to deduce from the
One-dimensional empirical measures, order statistics, and Kantorovich transport distances
This work is devoted to the study of rates of convergence of the empirical measures μn = 1 n ∑n k=1 δXk , n ≥ 1, over a sample (Xk)k≥1 of independent identically distributed real-valued random
On Modified Logarithmic Sobolev Inequalities for Bernoulli and Poisson Measures
We show that for any positive functionfon the discrete cube {0, 1}n,Entμnp(f)⩽pqEμnp1f|Df|2whereμnpis the product measure of the Bernoulli measure with probability of successp, as well as related
Extremal properties of half-spaces for log-concave distributions
n w x n where m s m = ??? = m is the product measure in R , D s y1, 1 is the n n-dimensional cube in R n and the infimum is then over all Borel-measurable n n . sets A ; R of measure m A G p; 0 - p -
Isoperimetric constants for product probability measures
A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived.
Weighted poincaré-type inequalities for cauchy and other convex measures
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