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116
Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
 J. FUNCT. ANAL
, 1999
"... 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 characterization of those probability measures which satisfy these inequalities. ..."
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Cited by 175 (9 self)
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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 characterization of those probability measures which satisfy these inequalities.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Fluctuations of eigenvalues and second order Poincaré inequalities
, 2007
"... Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified t ..."
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Cited by 52 (5 self)
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Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out; some of them are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.
Weighted Poincarétype inequalities for Cauchy and other convex measures
 ANNALS OF PROBABILITY
, 2007
"... Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, th ..."
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Cited by 30 (3 self)
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Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheegertype isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.
Discrete Isoperimetric And PoincaréType Inequalities
, 1996
"... We study some discrete isoperimetric and Poincar'etype inequalities for product probability measures ¯ n on the discrete cube f0; 1g n and on the lattice Z n . In particular we prove sharp lower estimates for the product measures of 'boundaries ' of arbitrary sets in the discre ..."
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Cited by 27 (2 self)
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We study some discrete isoperimetric and Poincar'etype inequalities for product probability measures ¯ n on the discrete cube f0; 1g n and on the lattice Z n . In particular we prove sharp lower estimates for the product measures of 'boundaries ' of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions ¯ on Z which satisfy these inequalities on Z n . The class of these distributions can be described by a certain class of monotone transforms of the twosided exponential measure. A similar characterization of distributions on R which satisfy Poincar'e inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. 1 Introduction For a number p 2 (0; 1), let ¯ p denote the Bernoulli measure on f0; 1g with probabilities q = 1 \Gamma p and p, respectively, and let ¯ n p denote the corresponding product measure on f0; 1g n . Denote by s i (x) the 'neighbour' of x 2 f0; 1g n obtained b...
Entropy jumps in the presence of a spectral gap
 Duke Math. J
, 2003
"... Abstract It is shown that if X is a random variable whose density satisfies a Poincar'e inequality, and Y is an independent copy of X, then the entropy of (X + Y)/p2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument us ..."
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Cited by 26 (5 self)
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Abstract It is shown that if X is a random variable whose density satisfies a Poincar'e inequality, and Y is an independent copy of X, then the entropy of (X + Y)/p2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of the BrunnMinkowski inequality (in its functional form due to Pr'ekopa and Leindler). 1 Introduction The informationtheoretic entropy of a real random variable X with density f: R! [0, 1) is defined as
Optimal Sobolev Imbeddings Involving RearrangementInvariant Quasinorms
, 1999
"... . Let m and n be positive integers with n 2 and 1 m n \Gamma 1. We study rearrangementinvariant quasinorms % R and % D on functions f : (0; 1) ! R such that to each bounded domain\Omega in R n , with Lebesgue measure j\Omega j, there corresponds C = C(j\Omega j) ? 0 for which one has the S ..."
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Cited by 24 (3 self)
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. Let m and n be positive integers with n 2 and 1 m n \Gamma 1. We study rearrangementinvariant quasinorms % R and % D on functions f : (0; 1) ! R such that to each bounded domain\Omega in R n , with Lebesgue measure j\Omega j, there corresponds C = C(j\Omega j) ? 0 for which one has the Sobolev imbedding inequality % R \Gamma u (j\Omega jt) \Delta C%D \Gamma jr m uj (j\Omega jt) \Delta ; u 2 C m 0 (\Omega\Gamma ; involving the nonincreasing rearrangements of u and a certain m th order gradient of u. When m = 1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which % D need not be rearrangementinvariant, % R \Gamma u (j\Omega jt) \Delta C%D ` d d t Z fx2R n : ju(x)j?u (j\Omega jt)g j(ru)(x)j dx ' ; u 2 C 1 0 : In both cases we are especially interested in when the quasinorms are optimal, in the sense that % R cannot be replaced by an essentially larger quasinorm and % D cannot be replaced by an essen...
Modified logarithmic Sobolev inequalities on R
, 2008
"... We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Götze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in ..."
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Cited by 18 (7 self)
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We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Götze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo. 1
On the role of convexity in functional and isoperimetric inequalities
 the Proc. London Math. Soc., arxiv.org/abs/0804.0453
, 2008
"... This is a continuation of our previous work [41]. It is well known that various isoperimetric inequalities imply their functional “counterparts”, but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for logconcave probability measures in Euclidean space) ..."
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Cited by 17 (6 self)
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This is a continuation of our previous work [41]. It is well known that various isoperimetric inequalities imply their functional “counterparts”, but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for logconcave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger’s inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, OrliczSobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz’ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no CentralLimit obstruction. As another application, we show that under our convexity assumptions, qlogSobolev inequalities (q ∈ [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvaturedimension condition of Bakry – Émery. 1
Functional inequalities for heavy tails distributions and application to isoperimetry
, 2008
"... Abstract. 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 Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and th ..."
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Cited by 16 (12 self)
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Abstract. 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 Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures onR n we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve many previous results.