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29
Hypercontractive measures, Talagrand’s inequality and influences
, 2011
"... Abstract. – We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and nonproduct models. The approach covers similarly by a simple interpolation the framework of geometric influences recent ..."
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Abstract. – We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and nonproduct models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric BrascampLieb decompositions are also considered in this context. 1.
An isoperimetric inequality on the ℓp balls
, 2008
"... The normalised volume measure on the ℓ n p unit ball (1 ≤ p ≤ 2) satisfies the following isoperimetric inequality: the boundary measure of a set of measure a is at least cn 1/p ã log 1−1/p (1/ã), where ã = min(a,1 − a). Résumé Nous prouvons une inégalité isopérimétrique pour la mesure uniforme Vp,n ..."
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The normalised volume measure on the ℓ n p unit ball (1 ≤ p ≤ 2) satisfies the following isoperimetric inequality: the boundary measure of a set of measure a is at least cn 1/p ã log 1−1/p (1/ã), where ã = min(a,1 − a). Résumé Nous prouvons une inégalité isopérimétrique pour la mesure uniforme Vp,n sur la boule unité de ℓ n p (1 ≤ p ≤ 2). Si Vp,n(A) = a, alors V + p,n(A) ≥ cn 1/p ea log 1−1/p 1/ea, où V + p,n est la mesure de surface associée à Vp,n, ea = min(a,1 − a) et c est une constante absolue. En particulier, les boules unités de ℓ n p vérifient la conjecture de Kannan– Lovász–Simonovits [KLS] sur la constante de Cheeger d’un corps convexe isotrope. La démonstration s’appuie sur les inégalités isopérimétriques de Bobkov [B1] et de Barthe–Cattiaux–Roberto [BCR], et utilise la représentation de Vp,n établie par Barthe–Guédon–Mendelson–Naor [BGMN] ainsi qu’un argument de découpage. 1
Bernstein type’s concentration inequalities for symmetric Markov processes
"... Abstract. Using the method of transportationinformation inequality introduced in [28],weestablishBernsteintype’sconcentrationinequalitiesforempiricalmeans 1 ∫ t t 0 g(Xs)ds where g is a unbounded observable of the symmetric Markov process (Xt). Three approaches are proposed: functional inequalities ..."
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Abstract. Using the method of transportationinformation inequality introduced in [28],weestablishBernsteintype’sconcentrationinequalitiesforempiricalmeans 1 ∫ t t 0 g(Xs)ds where g is a unbounded observable of the symmetric Markov process (Xt). Three approaches are proposed: functional inequalities approach; Lyapunov function method; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.
Weak logarithmic Sobolev inequalities and entropic convergence
, 2005
"... In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion ..."
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In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semigroup. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.
Poincaré inequalities for non euclidean metrics and . . .
, 2007
"... In this paper, we consider Poincaré inequalities for non euclidean metrics on R d. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian ..."
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In this paper, we consider Poincaré inequalities for non euclidean metrics on R d. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give different equivalent functional forms of these Poincaré type inequalities in terms of transportationcost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized BecknerLatalaOleszkiewicz inequalities.
Threshold Phenomena and Influence with some perspectives from Mathematics
 Computer Science and Economics
"... “Threshold phenomena ” refer to settings in which the probability for an event to occur changes rapidly as some underlying parameter varies. Threshold phenomena play an important role in probability theory and statistics, physics, and computer science, and are related to issues studied in economics ..."
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“Threshold phenomena ” refer to settings in which the probability for an event to occur changes rapidly as some underlying parameter varies. Threshold phenomena play an important role in probability theory and statistics, physics, and computer science, and are related to issues studied in economics
Convex Sobolev inequalities and spectral gap
, 2005
"... This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by ..."
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This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux [11] and Carlen and Loss [10] for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case.
AND
, 2006
"... Abstract. In this paper we derive non asymptotic deviation bounds for ∣∣ ∣ t ..."
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Abstract. In this paper we derive non asymptotic deviation bounds for ∣∣ ∣ t
Ecole Polytechnique and Université de Toulouse
, 2007
"... Abstract. This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound “à la Pinsker ” e ..."
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Abstract. This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound “à la Pinsker ” enabling us to study our problem firstly via usual functional inequalities (Poincaré inequality, weak Poincaré,...) and truncation procedure, and secondly through the introduction of new functional inequalities Iψ. These Iψinequalities are characterized through measurecapacity conditions and FSobolev inequalities. A direct study of the decay of Hellinger distance is also proposed. Finally we show how a dynamic approach based on reversing the role of the semigroup and the invariant measure can lead to interesting bounds. Résumé. Nous étudions ici la vitesse de convergence, pour la distance en variation totale, de diffusions ergodiques dont la loi initiale satisfait une intégrabilité donnée. Nous présentons différentes approches basées sur l’utilisation d’inégalités fonctionnelles. La première étape consiste à donner une borne générale à la Pinsker. Cette borne permet alors d’utiliser, en les combinant à une procedure de troncature, des inégalités usuelles (telles Poincaré ou Poincaré faibles,...). Dans un deuxième temps nous introduisons de nouvelles inégalités appelées Iψ que nous caractérisons à l’aide de condition de type capacitémesure et d’inégalités de type FSobolev. Une étude directe de la distance de Hellinger est également proposée. Pour conclure, une approche dynamique basée sur le renversement du rôle du semigroupe de diffusion et de la mesure invariante permet d’obtenir de nouvelles bornes intéressantes. Key words: total variation, diffusion processes, speed of convergence, Poincaré inequality, logarithmic Sobolev inequality, FSobolev inequality. MSC 2000: 26D10, 60E15.