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Generalization Of An Inequality By Talagrand, And Links With The Logarithmic Sobolev Inequality
 J. Funct. Anal
, 2000
"... . We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately logconcave, in a ..."
Abstract

Cited by 121 (10 self)
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. We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately logconcave, in a precise sense. All constants are independent of the dimension, and optimal in certain cases. The proofs are based on partial dierential equations, and an interpolation inequality involving the Wasserstein distance, the entropy functional and the Fisher information. Contents 1. Introduction 1 2. Main results 5 3. Heuristics 11 4. Proof of Theorem 1 18 5. Proof of Theorem 3 24 6. An application of Theorem 1 30 7. Linearizations 31 Appendix A. A nonlinear approximation argument 35 References 36 1. Introduction Let M be a smooth complete Riemannian manifold of dimension n, with the geodesic distance d(x; y) = inf 8 < : s Z 1 0 j _ w(t)j 2 dt; w 2 C 1 ((0; 1); M); w(0) = x; w(1) = y 9 ...
The BrunnMinkowski inequality
 Bull. Amer. Math. Soc. (N.S
, 2002
"... Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains ..."
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Cited by 74 (5 self)
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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the BrunnMinkowski inequality and other inequalities in geometry and analysis, and some applications. 1.
Concentration of the Spectral Measure for Large Matrices
, 2000
"... We derive concentration inequalities for functions of the empirical measure of eigenvalues for large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to nonGaussian Wigner and Wishart matrices. We also provide concentration bounds for ..."
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Cited by 65 (11 self)
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We derive concentration inequalities for functions of the empirical measure of eigenvalues for large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to nonGaussian Wigner and Wishart matrices. We also provide concentration bounds for non commutative functionals of random matrices. 1 Introduction and statement of results Consider a random N N Hermitian matrix X with i.i.d. complex entries (except for the symmetry constraint) satisfying a moment condition. It is well known since Wigner [28] that the spectral measure of N 1=2 X converges to the semicircle law. This observation has been generalized to a large class of matrices, e.g. sample covariance matrices of the form XRX where R is a deterministic diagonal matrix ([19]), band matrices (see [5, 16, 20]), etc. For the Wigner case, this convergence has been supplemented by Central Limit Theorems, see [15] for the case of Gaussian entries and [17], [22] for the gen...
Stochastic volatility models as hidden Markov models and statistical applications
 Bernoulli
, 2000
"... This paper deals with the ®xed sampling interval case for stochastic volatility models. We consider a twodimensional diffusion process (Yt, Vt), where only (Yt) is observed at n discrete times with regular sampling interval Ä. The unobserved coordinate (Vt) is ergodic and rules the diffusion coef®c ..."
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Cited by 45 (5 self)
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This paper deals with the ®xed sampling interval case for stochastic volatility models. We consider a twodimensional diffusion process (Yt, Vt), where only (Yt) is observed at n discrete times with regular sampling interval Ä. The unobserved coordinate (Vt) is ergodic and rules the diffusion coef®cient (volatility) of (Yt). We study the ergodicity and mixing properties of the observations (YiÄ). For this purpose, we ®rst present a thorough review of these properties for stationary diffusions. We then prove that our observations can be viewed as a hidden Markov model and inherit the mixing properties of (Vt). When the stochastic differential equation of (Vt) depends on unknown parameters, we derive momenttype estimators of all the parameters, and show almost sure convergence and a central limit theorem at rate n 1=2. Examples of models coming from ®nance are fully treated. We focus on the asymptotic variances of the estimators and establish some links with the small sampling interval case studied in previous papers.
Moment Inequalities for Functions of Independent Random Variables
"... this paper is to provide such generalpurpose inequalities. Our approach is based on a generalization of Ledoux's entropy method (see [26, 28]). Ledoux's method relies on abstract functional inequalities known as logarithmic Sobolev inequalities and provide a powerful tool for deriving exponential i ..."
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Cited by 39 (9 self)
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this paper is to provide such generalpurpose inequalities. Our approach is based on a generalization of Ledoux's entropy method (see [26, 28]). Ledoux's method relies on abstract functional inequalities known as logarithmic Sobolev inequalities and provide a powerful tool for deriving exponential inequalities for functions of independent random variables, see Boucheron, Massart, and AMS 1991 subject classifications. Primary 60E15, 60C05, 28A35; Secondary 05C80 Key words and phrases. Moment inequalities, Concentration inequalities; Empirical processes; Random graphs Supported by EU Working Group RANDAPX, binational PROCOPE Grant 05923XL The work of the third author was supported by the Spanish Ministry of Science and Technology and FEDER, grant BMF200303324 Lugosi [6, 7], Bousquet [8], Devroye [14], Massart [30, 31], Rio [36] for various applications. To derive moment inequalities for general functions of independent random variables, we elaborate on the pioneering work of Latala and Oleszkiewicz [25] and describe socalled #Sobolev inequalities which interpolate between Poincare's inequality and logarithmic Sobolev inequalities (see also Beckner [4] and Bobkov's arguments in [26])
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
Concentration inequalities
 Advanced Lectures in Machine Learning
, 2004
"... Abstract. Concentration inequalities deal with deviations of functions of independent random variables from their expectation. In the last decade new tools have been introduced making it possible to establish simple and powerful inequalities. These inequalities are at the heart of the mathematical a ..."
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Cited by 32 (1 self)
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Abstract. Concentration inequalities deal with deviations of functions of independent random variables from their expectation. In the last decade new tools have been introduced making it possible to establish simple and powerful inequalities. These inequalities are at the heart of the mathematical analysis of various problems in machine learning and made it possible to derive new efficient algorithms. This text attempts to summarize some of the basic tools. 1
The spectral gap for a Glaubertype dynamics in a continuous gas
, 2000
"... . We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenter ..."
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Cited by 26 (4 self)
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. We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenters are such that the Gibbs specication satises a certain mixing condition, then the spectral gap of the generator is strictly positive uniformly in the volume and boundary condition. The required mixing condition holds if, for instance, there is a convergent cluster expansion. Key Words: Spectral gap, Gibbs measures, continuous systems, birth and death processes Mathematics Subject Classication: 82C21, 60K35, 82C22, 60J75 This work was partially supported by GNAFA and by \Conanziamento Murst" v1.4 1. Introduction We consider a continuous gas in a bounded volume R d , distributed according the Gibbs probability measure associated to a nite range pair potential '. The Gibbs measu...
On Logarithmic Sobolev Inequalities For Continuous Time Random Walks On Graphs
 Probab. Theory Related Fields
, 2000
"... . We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard ..."
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Cited by 24 (2 self)
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. We establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals, such as the graph distance, in this setting. 1. Introduction The classical logarithmic Sobolev inequality for Brownian motion B = (B t ) t0 in IR d [Gr] indicates that for all functionals F in the domain of the Malliavin gradient operator D : L 2(\Omega ; IP) ! L 2(\Omega \Theta [0; T ]; IP\Omega dt), IE(F 2 log F 2 ) \Gamma IE(F 2 ) log IE(F 2 ) 2 IE `Z T 0 jD t F j 2 dt ' : (1:1) In particular, if F = f(B t 1 ; : : : ; B t n ), 0 = t 0 t 1 \Delta \Delta \Delta t n for some smooth function f : (IR d ) n ! IR, D t F = n X i=1 r i F I ft...
Concentration and Deviation Inequalities in Infinite Dimensions via Covariance Representations
, 2002
"... Concentration and deviation inequalities are obtained for functionals on Wiener space, Poisson space or more generally for normal martingales and binomial processes. The method used here is based on covariance identities obtained via the chaotic representation property, and provides an alternative ..."
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Cited by 22 (11 self)
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Concentration and deviation inequalities are obtained for functionals on Wiener space, Poisson space or more generally for normal martingales and binomial processes. The method used here is based on covariance identities obtained via the chaotic representation property, and provides an alternative to the use of logarithmic Sobolev inequalities. It allows to recover known concentration and deviation inequalities on the Wiener and Poisson space (including the ones given by sharp logarithmic Sobolev inequalities), and extends results available in the discrete case, i.e. on the infinite cube {1; 1}∞.