# Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry

@article{Cattiaux2008FunctionalIF, 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}, year={2008}, volume={15}, pages={346-385} }

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…

## 54 Citations

### Weak Poincaré inequalities for convolution probabilities measures

- MathematicsInfinite Dimensional Analysis, Quantum Probability and Related Topics
- 2019

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

- MathematicsBernoulli
- 2019

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

- Mathematics
- 2020

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…

### ON SPECTRAL GAP AND WEIGHTED POINCARÉ INEQUALITIES FOR SOME ONE-DIMENSIONAL DIFFUSIONS

- Mathematics
- 2015

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

- MathematicsQuaestiones Mathematicae
- 2022

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…

### Quantitative isoperimetric inequalities for log-convex probability measures on the line

- Mathematics
- 2014

### On the Poincaré Constant of Log-Concave Measures

- MathematicsLecture Notes in Mathematics
- 2020

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…

### A NOTE ON SPECTRAL GAP AND WEIGHTED POINCAR ´ E INEQUALITIES FOR SOME ONE-DIMENSIONAL DIFFUSIONS

- Mathematics
- 2014

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

- MathematicsAnnales Henri Poincaré
- 2019

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

- Mathematics
- 2010

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…

## References

SHOWING 1-10 OF 113 REFERENCES

### Weighted poincaré-type inequalities for cauchy and other convex measures

- Mathematics
- 2009

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

- Mathematics
- 2010

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

- Mathematics
- 2010

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…

### Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups

- Mathematics
- 2001

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

- Mathematics
- 1999

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

- Mathematics
- 2005

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

- Mathematics
- 2007

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

- Mathematics
- 2003

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…