Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry

  title={Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry},
  author={Patrick Cattiaux and Nathael Gozlan and Arnaud Guillin and Cyril Roberto},
  journal={Electronic Journal of Probability},
This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincare and weak Cheeger, weighted Poincare and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $\mathbb{R}^n$ we obtain the optimal dimension dependence using the mass transportation method. Then we derive… 

Weak Poincaré inequalities for convolution probabilities measures

In this paper, weak Poincaré inequalities are obtained for convolution probabilities with explicit rate functions by constructing suitable Lyapunov functions. Here, one of these Lyapunov functions is

Weighted Poincaré inequalities, concentration inequalities and tail bounds related to Stein kernels in dimension one

We investigate links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a

Fokker-Planck equations and one-dimensional functional inequalities for heavy tailed densities

We study one-dimensional functional inequalities of the type of Poincare, logarithmic Sobolev and Wirtinger, with weight, for probability densities with polynomial tails. As main examples, we obtain


We present some classical and weighted Poincaré inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors for

Variations and extensions of the Gaussian concentration inequality, Part I

We use and modify the Gaussian concentration inequality to prove a variety of concentration inequalities for a wide class of functions and measures on $\mathbb{R}^{n}$, typically involving

On the Poincaré Constant of Log-Concave Measures

The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincare constant. First we revisit E. Milman's result [40] on the link


We present some classical and weighted Poincar\'e inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors

Weak Poincaré Inequalities in the Absence of Spectral Gaps

For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called weak Poincaré inequality (WPI), originally introduced by

Bernstein type's concentration inequalities for symmetric Markov processes

Using the method of transportation-information inequality introduced in [A. Guillin et al., Probab. Theory Related Fields, 144 (2009), pp. 669--695], we establish Bernstein-type concentration



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

Poincaré inequalities and dimension free concentration of measure

In this paper, we consider Poincare inequalities for non euclidean metrics on $\mathbb{R}^d$. These inequalities enable us to derive precise dimension free concentration inequalities for product

Distributions with Slow Tails and Ergodicity of Markov Semigroups in Infinite Dimensions

We discuss some geometric and analytic properties of probability distributions that are related to the concept of weak Poincare type inequalities. We deal with isoperimetric and capacitary

Lyapunov conditions for Super Poincaré inequalities

Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups

Abstract In order to describe L 2 -convergence rates slower than exponential, the weak Poincare inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the

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

Entropy Bounds and Isoperimetry

Introduction and notations Poincare-type inequalities Entropy and Orlicz spaces $\mathbf{LS}_q$ and Hardy-type inequalities on the line Probability measures satisfying $\mathbf{LS}_q$-inequalities on

Lyapunov conditions for logarithmic Sobolev and Super Poincar\'e inequality

We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincar\'e inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincar\'e and

Sobolev inequalities for probability measures on the real line

We give a characterization of those probability measures on the real line which satisfy certain Sobolev inequalities. Our starting point is a simpler approach to the Bobkov–Götze characterization of