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29 results on '"Michel Ledoux"'

1. On Optimal Matching of Gaussian Samples

Author

Michel Ledoux

Subjects
Statistics and Probability, Conjecture, Applied Mathematics, General Mathematics, Gaussian, 010102 general mathematics, Order (ring theory), Gaussian measure, 01 natural sciences, 010305 fluids & plasmas, Combinatorics, symbols.namesake, Common distribution, 0103 physical sciences, symbols, 0101 mathematics, Mass transportation, Random variable, Mathematics
Abstract

Let X1, . . .,Xn be independent random variables having as common distribution the standard Gaussian measure μ on ℝ2 and let $$ {\mu}_n=\frac{1}{n}\sum \limits_{i=1}^n{\delta}_{X_i} $$ be the associated empirical measure. We show that $$ \frac{1}{C}\frac{\log n}{n}\le $$ 𝔼 $$ \left({\mathrm{W}}_2^2\left({\mu}_n,\mu \right)\right)\le C\frac{{\left(\log n\right)}^2}{n} $$ for some numerical constant C > 0, where W2 is the quadratic Kantorovich metric, and conjecture that the left-hand side provides the correct order. The proof is based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra, and D. Trevisan.

Published
2019
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2. Transport Inequalities on Euclidean Spaces for Non-Euclidean Metrics

Author

Michel Ledoux and Sergey G. Bobkov

Subjects
Optimal matching, Partial differential equation, Applied Mathematics, General Mathematics, 010102 general mathematics, 020206 networking & telecommunications, 02 engineering and technology, 01 natural sciences, Algebra, symbols.namesake, Fourier analysis, Non-Euclidean geometry, Euclidean geometry, 0202 electrical engineering, electronic engineering, information engineering, symbols, 0101 mathematics, Focus (optics), Real line, Analysis, Mathematics, Probability measure
Abstract

We explore upper bounds on Kantorovich transport distances between probability measures on the Euclidean spaces in terms of their Fourier-Stieltjes transforms, with focus on non-Euclidean metrics. The results are illustrated on empirical measures in the optimal matching problem on the real line.

Published
2020
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3. A Law of the Iterated Logarithm for Directed Last Passage Percolation

Author

Michel Ledoux

Subjects
Statistics and Probability, Discrete mathematics, Superadditivity, General Mathematics, 010102 general mathematics, Law of the iterated logarithm, 01 natural sciences, Directed percolation, Combinatorics, Iterated logarithm, 010104 statistics & probability, Tracy–Widom distribution, 0101 mathematics, Statistics, Probability and Uncertainty, Random variable, Mathematics
Abstract

Let $${\widetilde{H}}_N$$ , $$N \ge 1$$ , be the point-to-point last passage times of directed percolation on rectangles $$[(1,1), ([\gamma N], N)]$$ in $${\mathbb {N}}\times {\mathbb {N}}$$ over exponential or geometric independent random variables, rescaled to converge to the Tracy–Widom distribution. It is proved that for some $$\alpha _{\sup } >0$$ , $$\begin{aligned} \alpha _{\sup } \, \le \, \limsup _{N \rightarrow \infty } \frac{{\widetilde{H}}_N}{(\log \log N)^{2/3}} \, \le \, \Big ( \frac{3}{4} \Big )^{2/3} \end{aligned}$$ with probability one, and that $$\alpha _{\sup } = \big ( \frac{3}{4} \big )^{2/3}$$ provided a commonly believed tail bound holds. The result is in contrast with the normalization $$(\log N)^{2/3}$$ for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm. A weaker result on the liminf with speed $$(\log \log N)^{1/3}$$ is also discussed.

Published
2017
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4. A Dimension-Free Reverse Logarithmic Sobolev Inequality for Low-Complexity Functions in Gaussian Space

Author

Michel Ledoux and Ronen Eldan

Subjects
Mathematics::Functional Analysis, Pure mathematics, Gaussian, 010102 general mathematics, Mathematical proof, Gaussian measure, Space (mathematics), 01 natural sciences, Low complexity, 010104 statistics & probability, symbols.namesake, Dimension (vector space), symbols, 0101 mathematics, GEOM, Mathematics, Logarithmic sobolev inequality
Abstract

We discuss new proofs, and new forms, of a reverse logarithmic Sobolev inequality, with respect to the standard Gaussian measure, for low-complexity functions, measured in terms of Gaussian-width. In particular, we provide a dimension-free improvement for a related result given in Eldan (Geom Funct Anal 28:1548–1596, 2018).

Published
2020
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5. Heat Flow Derivatives and Minimum Mean-Square Error in Gaussian Noise

Author

Michel Ledoux

Subjects
Minimum mean square error, 010102 general mathematics, Orthogonality principle, 020206 networking & telecommunications, 02 engineering and technology, Library and Information Sciences, 01 natural sciences, Computer Science Applications, Combinatorics, symbols.namesake, Additive white Gaussian noise, Gaussian noise, Iterated function, 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, Entropy (information theory), 0101 mathematics, Fisher information, Random variable, Information Systems, Mathematics
Abstract

We connect recent developments on Gaussian noise estimation and the Minimum Mean-Square Error to earlier results on entropy and Fisher information heat flow expansion. In particular, the derivatives of the Minimum mean-square error with respect to the noise parameter are related to the heat flow derivatives of the Fisher information and a special Lie algebra structure on iterated gradients. The results lead in particular to a partial answer to the Minimum mean-square error conjecture.

Published
2016
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6. Mass transportation proofs of free functional inequalities, and free Poincaré inequalities

Author

Ionel Popescu, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Georgia Institute of Technology [Atlanta], Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mass transport, Pure mathematics, Spectral gap, Distribution (number theory), Context (language use), Mathematical proof, 01 natural sciences, 010104 statistics & probability, symbols.namesake, FOS: Mathematics, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, Probability (math.PR), 010102 general mathematics, Functional inequalities, Free probability, Functional Analysis (math.FA), Mathematics - Functional Analysis, Poincaré conjecture, symbols, Random matrices, Convex function, Random matrix, Mathematics - Probability, Analysis
Abstract

This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincar\'e inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on $\R_{+}$ with the reference example of the Marcenko-Pastur distribution., Comment: This paper will apear in Journal of Fucntional Analysis

Published
2009
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7. A general framework for simulation of fractional fields

Author

Michel Ledoux, Serge Cohen, Céline Lacaux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Fractional Brownian motion, Random field, Series (mathematics), Stochastic process, Applied Mathematics, 010102 general mathematics, Mathematical analysis, 01 natural sciences, Measure (mathematics), Fractional fields, Fractional calculus, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Random measure, Convergence of random variables, Modeling and Simulation, Modelling and Simulation, Infinitely divisible distributions, 0101 mathematics, Simulation of random fields, Mathematics
Abstract

International audience; Besides fractional Brownian motion most non-Gaussian fractional fields are obtained by integration of deterministic kernels with respect to a random infinitely divisible measure. In this paper, generalized shot noise series are used to obtain approximations of most of these fractional fields, including linear and harmonizable fractional stable fields. Almost sure and $L^r$-norm rates of convergence, relying on asymptotic developments of the deterministic kernels, are presented as a consequence of an approximation result concerning series of symmetric random variables. When the control measure is infinite, normal approximation has to be used as a complement. The general framework is illustrated by simulations of classical fractional fields.

Published
2008
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8. A Stein deficit for the logarithmic Sobolev inequality

Author

Giovanni Peccati, Ivan Nourdin, and Michel Ledoux

Subjects
Pure mathematics, Semigroup, General Mathematics, Gaussian, 010102 general mathematics, Probability (math.PR), 020206 networking & telecommunications, 02 engineering and technology, Characterization (mathematics), Information theory, 01 natural sciences, Functional Analysis (math.FA), Mathematics - Functional Analysis, symbols.namesake, Distribution (mathematics), 0202 electrical engineering, electronic engineering, information engineering, symbols, FOS: Mathematics, 0101 mathematics, Representation (mathematics), Fisher information, Mathematics - Probability, Logarithmic sobolev inequality, Mathematics
Abstract

We provide explicit lower bounds for the deficit in the Gaussian logarithmic Sobolev inequality in terms of differential operators that are naturally associated with the so-called Stein characterization of the Gaussian distribution. The techniques are based on a crucial use of the representation of the relative Fisher information, along the Ornstein-Uhlenbeck semigroup, in terms of the Minimal Mean-Square Error from information theory., Comment: 22 pages

Published
2016
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9. A logarithmic Sobolev form of the Li-Yau parabolic inequality

Author

Dominique Bakry, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
60H, 60J, General Mathematics, non-negative curvature, Poincaré inequality, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, heat semigroup, gradient estimate, Sobolev inequality, Li-Yau inequality, 010104 statistics & probability, symbols.namesake, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Log sum inequality, diameter bound, 0101 mathematics, logarithmic Sobolev inequality, Heat kernel, Mathematics, Markov semigroups, heat equation, Euclidean space, 010102 general mathematics, Mathematical analysis, crvature-dimension inequalities, Gaussian measure, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Sobolev space, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Li-Yau parabolic inequality, symbols, Heat equation, 58J, diffusion semigroups
Abstract

International audience; We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities in this setting. Exponential Laplace differential inequalities through the Herbst argument furthermore yield diameter bounds and dimensional estimates on the heat kernel volume of balls.

Published
2006
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10. Quantitative logarithmic Sobolev inequalities and stability estimates

Author

Max Fathi, Michel Ledoux, Emanuel Indrei, Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Poincaré inequality, Context (language use), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Upper and lower bounds, Sobolev inequality, 010104 statistics & probability, symbols.namesake, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, Discrete Mathematics and Combinatorics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Heat kernel, Mathematics, Semigroup, Applied Mathematics, 010102 general mathematics, Probability (math.PR), Gaussian measure, Functional Analysis (math.FA), Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Metric (mathematics), symbols, Analysis, Mathematics - Probability
Abstract

International audience; We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\'e inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${\rm L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-\'Emery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.

Published
2014
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11. FROM LOG SOBOLEV TO TALAGRAND: A QUICK PROOF

Author

Michel Ledoux, Nicola Gigli, Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, log-Sobolev inequality, Space (mathematics), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Measure (mathematics), Metric measure spaces, Sobolev inequality, 010104 statistics & probability, symbols.namesake, Development (topology), Settore MAT/05 - Analisi Matematica, Mathematics::Metric Geometry, Discrete Mathematics and Combinatorics, functional inequalities, 0101 mathematics, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics, Applied Mathematics, 010102 general mathematics, Hilbert space, optimal transport, Talagrand inequality, Sobolev space, Poincaré conjecture, Metric (mathematics), symbols, Analysis
Abstract

International audience; We provide yet another proof of the Otto-Villani theorem from the log Sobolev inequality to the Talagrand transportation cost inequality valid in arbitrary metric measure spaces. The argument relies on the recent development [2] identifying gradient flows in Hilbert space and in Wassertein space, emphasizing one key step as precisely the root of the Otto-Villani theorem. The approach does not require the doubling property or the validity of the local Poincare' inequality.

Published
2013

12. The one dimensional free Poincaré inequality

Author

Michel Ledoux, Ionel Popescu, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [ Georgia Institute of Technology], Georgia Institute of Technology [Atlanta], 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Chebyshev polynomials, Dirichlet form, Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematics - Operator Algebras, Poincaré inequality, 16. Peace & justice, Free probability, Multidimensional Chebyshev's inequality, Chebyshev's sum inequality, 01 natural sciences, Mathematics - Functional Analysis, 010104 statistics & probability, symbols.namesake, symbols, 46L54 (33D45 60B20 60E15), Almost surely, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, Probability measure
Abstract

In this paper we discuss the natural candidate for the one dimensional free Poincar\'e inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional inequalities, namely, the free transportation and Log-Sobolev inequalities. As in the classical case the Poincar\'e is implied by the others. This investigation is driven by a nice lemma of Haagerup which relates logarithmic potentials and Chebyshev polynomials. The Poincar\'e inequality revolves around the counting number operator for the Chebyshev polynomials of first kind with respect to the arcsine law on $[-2,2]$. This counting number operator appears naturally in a representation of the minimum of the logarithmic potential with external fields as well as in the perturbation of logarithmic energy with external fields, which is the essential connection between all these inequalities., Comment: Some things were corrected and a new proposition added. This will appear in Transactions of AM

Published
2013
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13. Chaos of a Markov operator and the fourth moment condition

Author

Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, iterated gradient, Gaussian, Stein’s method, Markov operator, 01 natural sciences, 010104 statistics & probability, symbols.namesake, Mathematics::Probability, 60F05, Convergence (routing), Gamma distribution, FOS: Mathematics, Applied mathematics, fourth moment, eigenfunction, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, 60J60, Sequence, $\Gamma$-calculus, 010102 general mathematics, Spectrum (functional analysis), Probability (math.PR), Eigenfunction, 16. Peace & justice, Distribution (mathematics), Iterated function, symbols, 60J35, Chaos, 60F05 (60H99 60J35 60J99), 60J99, Statistics, Probability and Uncertainty, 60H99, Mathematics - Probability
Abstract

We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we are led to introduce a notion of chaos associated to a Markov operator through its iterated gradients and present conditions on the (pure) point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussian distribution. Convergence to gamma distributions may be examined similarly., Comment: Published in at http://dx.doi.org/10.1214/11-AOP685 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2012
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14. Hypercontractive measures, Talagrand's inequality, and influences

Author

Dario Cordero-Erausquin, Michel Ledoux, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Mathematics::Functional Analysis, Cayley graph, Inequality, Dirichlet form, media_common.quotation_subject, 010102 general mathematics, 60E15 (28A12), Context (language use), Type (model theory), 01 natural sciences, 010104 statistics & probability, Simple (abstract algebra), Product (mathematics), 0101 mathematics, Mathematical economics, ComputingMilieux_MISCELLANEOUS, Mathematics, media_common, Interpolation
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 non-product 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 Brascamp-Lieb decompositions are also considered in this context.

Published
2012
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15. From concentration to isoperimetry : semigroup proofs

Author

Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, 60E15 (26D10 31C15 46E35 58J65), Series (mathematics), Semigroup, 010102 general mathematics, 01 natural sciences, Measure (mathematics), 010104 statistics & probability, Geometric measure theory, Simple (abstract algebra), Cover (algebra), 0101 mathematics, Isoperimetric inequality, Heat kernel, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

In a remarkable series of works, E. Milman recently showed how to reverse the usual hierarchy, and deduce from measure concentration inequali- ties, dimension free isoperimetric type inequalities in spaces with non-negative (Ricci) curvature. The results cover two basic instances, linear isoperimetry under arbitrarily slow concentration, and logarithmic strengthenings above the linear case under exponential concentration. The proofs are developed in a Rie- mannian (with density) context making use of isoperimetric minimizers and refined tools from geometric measure theory. In this note, we present simple semigroup arguments to cover the super-linear case, of potential usefulness in more general settings. A particular emphasis is put on functional inequalities for heat kernel measures.

Published
2011
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16. Correlation and Brascamp-Lieb Inequalities for Markov Semigroups

Author

Franck Barthe, Bernard Maurey, Dario Cordero-Erausquin, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Superadditivity, Inequality, Markov chain, Euclidean space, Semigroup, General Mathematics, media_common.quotation_subject, Probability (math.PR), 010102 general mathematics, Mathematical proof, 01 natural sciences, Functional Analysis (math.FA), Algebra, Mathematics - Functional Analysis, 010104 statistics & probability, Symmetric group, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, Entropy (information theory), 0101 mathematics, Mathematics - Probability, Mathematics, media_common
Abstract

International audience; This paper builds upon several recent works, where semigroup proofs of Brascamp-Lieb inequalities are provided in various settings (Euclidean space, spheres, and symmetric groups). Our aim is two-fold. Firstly, we provide a general, unifying, framework based on Markov generators, in order to cover a variety of examples of interest going beyond previous investigations. Secondly, we put forward the combinatorial reasons for which unexpected exponents occur in these inequalities. Related superadditivity of information and entropy inequalities are also studied.

Published
2011
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17. Small deviations for beta ensembles

Author

Michel Ledoux, Brian Rider, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Mathematics::Classical Analysis and ODEs, 01 natural sciences, Combinatorics, 010104 statistics & probability, FOS: Mathematics, Order (group theory), Beta (velocity), Statistical physics, 0101 mathematics, [MATH]Mathematics [math], Eigenvalues and eigenvectors, ComputingMilieux_MISCELLANEOUS, Mathematics, 60B20, 010102 general mathematics, Small deviations, Probability (math.PR), eigenvalues, Variance (accounting), Mathematics::Spectral Theory, small deviations, Statistics, Probability and Uncertainty, Random matrices, Random matrix, 60F99, Mathematics - Probability, 60B20 (60F99)
Abstract

International audience; We establish various small deviation inequalities for the extremal (soft edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both settings, upper bounds on the variance of the largest eigenvalue of the anticipated order follow immediately.

Published
2009
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18. A recursion formula for the moments of the Gaussian orthogonal ensemble

Author

Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Largest eigenvalue, Differential equation, Small deviation inequality, Gaussian, 01 natural sciences, Combinatorics, 010104 statistics & probability, symbols.namesake, Probability theory, Moment recursion formula, 60E05, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Eigenvalues and eigenvectors, Mathematics, Map enumeration, Recurrence relation, Laplace transform, 46L54, 010102 general mathematics, 82B31, Recursion (computer science), 15A52, 33C45, symbols, Statistics, Probability and Uncertainty, Gaussian Orthogonal Ensemble
Abstract

Ce travail presente un analogue de la relation de recurrence de Harer et Zagier pour les moments de l’Ensemble Orthogonal Gaussien sous la forme d’une recurrence a cinq termes. La demonstration s’appuie sur des integrations par parties gaussiennes et des equations differentielles sur les transformees de Laplace. Une relation similaire est etablie pour l’Ensemble Symplectique Gaussien. Comme dans le cas complexe, cette relation s’interprete comme une formule de recurrence pour le nombre de cartes enracinees a nombre de faces et de cotes donne plongees dans des surfaces localement orientees. Cette relation de recurrence sur les moments fournit egalement une borne sur la loi de la plus grande valeur propre de l’Ensemble Orthogonal Gaussien et, par comparaison de moments, de familles de matrices de Wigner.

Published
2009
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19. Weighted Poincaré-type inequalities for Cauchy and other convex measures

Author

Sergey G. Bobkov, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, 60B11, Brascamp–Lieb-type inequalities, Type (model theory), 01 natural sciences, Measure (mathematics), infimum convolution, 010104 statistics & probability, Probability theory, 60G07, 46G12 (60B11 60G07), Applied mathematics, 46G12, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Probability measure, Mathematics, logarithmic Sovolev inequalities, measure concentration, Hierarchy (mathematics), 010102 general mathematics, Regular polygon, Cauchy distribution, 46G12, 60B11, 60G07 (Primary), Cheeger-type inequalities, Statistics, Probability and Uncertainty, Isoperimetric inequality, weighted Poincaré-type inequalities, Mathematics - Probability
Abstract

Brascamp--Lieb-type, weighted Poincar\'{e}-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general $\kappa$-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration., Comment: Published in at http://dx.doi.org/10.1214/08-AOP407 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2009
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20. The geometry of Euclidean convolution inequalities and entropy

Author

Dario Cordero-Erausquin, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Young's inequality, Hermite polynomials, Applied Mathematics, General Mathematics, 010102 general mathematics, Probability (math.PR), 42A85 (52A40 60E15 60G15 94A17), Geometry, Convolution power, Information theory, 01 natural sciences, Functional Analysis (math.FA), Mathematics - Functional Analysis, Harmonic analysis, 010104 statistics & probability, Euclidean geometry, FOS: Mathematics, Entropy (information theory), 0101 mathematics, Convolution theorem, Mathematics - Probability, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

The goal of this note is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of $\R^2$. We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.

Published
2009
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21. An observation about submatrices

Author

Sourav Chatterjee, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, 01 natural sciences, 60E15 (15A18 15B52 60B20), Combinatorics, 010104 statistics & probability, Symmetric group, FOS: Mathematics, eigenvalue, 0101 mathematics, Real line, ComputingMilieux_MISCELLANEOUS, Eigenvalues and eigenvectors, Mathematics, empirical distribution, Probability (math.PR), 010102 general mathematics, 15A52, Block matrix, 60E15, 15A52, Random matrix, Random walk, Hermitian matrix, concentration of measure, Singular value, 60E15, Statistics, Probability and Uncertainty, Mathematics - Probability
Abstract

Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of M of order k. The proof uses results about random walks on symmetric groups and concentration of measure. In a similar way, we also show that almost all k x n submatrices of M have almost the same distribution of singular values., 6 pages

Published
2008

22. From Brunn-Minkowski to sharp Sobolev inequalities

Author

Michel Ledoux, Sergey G. Bobkov, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics::Functional Analysis, Pure mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, 46E35 (26D15), Mathematics::General Topology, Poincaré inequality, Mathematics::Algebraic Topology, 01 natural sciences, Sobolev inequality, 010101 applied mathematics, symbols.namesake, Simple (abstract algebra), Minkowski space, symbols, Mathematics::Metric Geometry, Log sum inequality, Direct proof, Rearrangement inequality, 0101 mathematics, Constant (mathematics), ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We present a simple direct proof of the classical Sobolev inequality in \(\mathbb{R}^n\) with best constant from the geometric Brunn–Minkowski–Lusternik inequality.

Published
2008
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23. Perturbations of functional inequalities using growth conditions

Author

Dominique Bakry, Feng-Yu Wang, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), School of Mathematical Sciences [Beijing] (SMS), Beijing Normal University (BNU), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics(all), Pure mathematics, Inequality, Applied Mathematics, General Mathematics, media_common.quotation_subject, Logarithmic Sobolev inequality, 010102 general mathematics, Mathematical analysis, Poincaré inequality, 60J45 (58J65 60E15), 01 natural sciences, Perturbation, 010104 statistics & probability, symbols.namesake, symbols, 0101 mathematics, Reference function, ComputingMilieux_MISCELLANEOUS, Logarithmic sobolev inequality, Mathematics, media_common
Abstract

Perturbations of functional inequalities are studied by using merely growth conditions in terms of a distance-like reference function. As a result, optimal sufficient conditions are obtained for perturbations to reach a class of functional inequalities interpolating between the Poincaré inequality and the logarithmic Sobolev inequality.

Published
2007
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24. On measure concentration of vector-valued maps

Author

Krzysztof Oleszkiewicz, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Pure mathematics, General Computer Science, Concentration of measure, Gaussian, 010102 general mathematics, Gaussian measure, Lipschitz continuity, 01 natural sciences, Measure (mathematics), 010104 statistics & probability, symbols.namesake, symbols, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, 60E15 (60G15), Mathematics
Abstract

In this work, we study concentration properties for vector valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in Rk. To this task, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions to earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite dimensional setting.

Published
2007
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25. Deviation inequalities on largest eigenvalues

Author

Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Inequality, media_common.quotation_subject, Gaussian, 010102 general mathematics, 01 natural sciences, Measure (mathematics), Exponential function, 010104 statistics & probability, symbols.namesake, 60E15 (15A42 60K35 82B31), symbols, Applied mathematics, Simplicity, 0101 mathematics, Concentration inequality, Random matrix, Eigenvalues and eigenvectors, media_common, Mathematics
Abstract

In these notes, we survey developments on the asymptotic behavior of the largest eigenvalues of random matrix and random growth models, and describe the corresponding known non-asymptotic exponential bounds. We then discuss some elementary and accessible tools from measure concentration and functional analysis to reach some of these quantitative inequalities at the correct small deviation rate of the fluctuation theorems. Results in this direction are rather fragmentary. For simplicity, we mostly restrict ourselves to Gaussian models.

Published
2007
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26. More hypercontractive bounds for deformed orthogonal polynomial ensembles

Author

Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Combinatorics, Wishart distribution, 010104 statistics & probability, 010102 general mathematics, Orthogonal polynomials, 60F15 (15A52 62E15), 0101 mathematics, 01 natural sciences, Mathematics
Published
2007
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27. Séminaire de Probabilités XXXVI

Author

Michel Émery, Marc Yor, Jacques Azéma, Michel Ledoux, and Séminaire de probabilités

Subjects
Pure mathematics, 010102 general mathematics, Mathematical analysis, Vector bundle, 01 natural sciences, Fock space, Sobolev inequality, Sobolev space, 010104 statistics & probability, Stochastic differential equation, Mathematics::Probability, Laguerre polynomials, 0101 mathematics, Martingale (probability theory), Random matrix, Mathematics
Abstract

Cours specialises et exposes thematiques: A. Guionnet, B. Zegarlinski: Lectures on Logarithmic Sobolev inequalities.- A. Lejay, L. Pastur: Matrices aleatoires : statistique asymptotique des valeurs propres.- N. O'Connell: Random matrices, non-colliding particle systems and queues.- Exposes: A. Dermoune, O. Moutsinga: Generalized variational principles.- D. Chafai: Gaussian maximum of entropy and reversed log-Sobolev inequality.- L. Miclo: About projections of logarithmic Sobolev inequalities.- L. Miclo: Sur l'inegalite de Sobolev logarithmique des operateurs de Laguerre a petit parametre.- A. Bentaleb: Sur les fonctions extremales des inegalites de Sobolev des operateurs de diffusion.- C. Donati-Martin, Y. Hu: Penalization of the Wiener measure and principal values.- C. Leuridan: Theoreme de Ray-Knight dans un arbre : Une approche algebrique.- R. Bass: Stochastic differential equations driven by symmetric stable processes.- T. Simon: Support d'une equation d' Ito avec sauts en dimension 1.- N. Eisenbaum: A Gaussian sheet connected to symmetric Markov chains.- C. Leuridan: Filtration d'une marche aleatoire stationnaire sur le cercle.- S. Beghdadi-Sakrani: Une martingale non pure, dont la filtration est brownienne.- J. Hannig: On filtrations related to purely discontinuous martingales.- S. Beghdadi-Sakrani: Calcul stochastique pour des mesures signees.- J. Jacod: On processes with conditional independent increments and stable convergence in law.- V. Grecea: Duality and quasi-continuity for supermartingales.- Y. Kabanov, C. Stricker: On the true submartingale property, d'apres Schachermayer.- C. Stricker: Simple strategies in exponential utility maximization.- M. Arnaudon, A. Thalmaier: Horizontal martingales in vector bundles.- D. Kurtz: Representation nucleaire des martingales d'Azema.- S. Attal: Approximating the Fock space with the toy Fock space.- Corrections auxvolumes anterieurs.

Published
2003
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28. Paneitz-type operators and applications

Author

Emmanuel Hebey, Zindine Djadli, Michel Ledoux, Analyse, Géométrie et Modélisation (AGM - UMR 8088), CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS), and Abdelmoumene, Amina

Subjects
010101 applied mathematics, Algebra, General Mathematics, 010102 general mathematics, 58J60, 0101 mathematics, Type (model theory), 35B45, 01 natural sciences, Mathematics
Published
2000

29. Méthodes fonctionnelles pour des grandes déviations quasi-gaussiennes

Author

Michel Ledoux, Djalil Chafaï, Laboratoire de Statistique et Probabilités (LSP), Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Semigroup, Gaussian, 010102 general mathematics, Mathematical analysis, Banach space, General Medicine, 01 natural sciences, Upper and lower bounds, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, symbols.namesake, Probability theory, Boltzmann constant, symbols, Applied mathematics, Large deviations theory, 0101 mathematics, Hamiltonian (quantum mechanics), Mathematics
Abstract

International audience; Some Gaussian functional inequalities have simple generalizations to some Gaussianlike cases. They allow us to establish Gaussian-like Large Deviations Principles and bounds via Gaussian concentration and shift inequalities for certain families of Boltzmann measures and laws of diffusion semigroups in short time. Beyond the results themselves, we would like to emphasize here the method and the symmetry of the arguments used for upper and lower bounds by means of the functional inequalities.

Published
1999
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