2013

Article in peer-reviewed journal

Titre
Processus des restaurants chinois et loi d'Ewens
Auteurs
Djalil Chafaï ; Yan Doumerc; Florent Malrieu
Détail
Revue de Mathématiques Spéciales (RMS), 2013, 123 (3), pp. 10-20
Début du résumé
On étudie une suite aléatoire à valeurs dans les permutations d'ensembles finis, appelée processus des restaurants chinois. Ce processus est relié à la loi d'Ewens bien connue en combinatoire élémentaire. Ce processus et cette loi constituent en quelque sorte un analogue pour les permutations du processus de Poisson et de la loi de Poisson, plus classiques en théorie des probabilités. .....
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Preprint, Working Paper, ...

Titre
First order global asymptotics for Calogero-Sutherland gases
Auteurs
Djalil Chafaï ; Nathael Gozlan; Pierre-André Zitt
Détail
Apr. 2013
Début du résumé
We study a physical system of N interacting particles in Rd, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as N tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension d>2, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as N tends to infinity. In the more specific case of Coulomb interaction in dimension d>2, and .....
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Titre
Circular law for random matrices with unconditional log-concave distribution
Auteurs
Détail
Mar. 2013
Début du résumé
We explore the validity of the circular law for random matrices with non i.i.d. entries. Let A be a random n × n real matrix having as a random vector in R^{n^2} a log-concave isotropic unconditional law. In particular, the entries are uncorellated and have a symmetric law of zero mean and unit variance. This allows for some dependence and non equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries. Our main result states that as n goes to infinity, the empirical spectral distribution of n^{-1/2}A tends to the uniform law on the unit disc .....
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2012

Article in peer-reviewed journal

Titre
Circular Law Theorem for Random Markov Matrices
Auteurs
Charles Bordenave; Pietro Caputo; Djalil Chafai
Détail
Probability Theory and Related Fields, 2012, 152 (3-4), pp. 751-779
DOI
DOI : 10.1007/s00440-010-0336-1
Début du résumé
Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded density, mean m, and finite positive variance sigma^2. Let M be the nxn random Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its sum. In particular, when X11 follows an exponential law, then M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Our main result states that with probability one, the counting probability measure of the complex spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law on the centered disk of radius sigma/m. The .....
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Titre
Around the circular law
Auteurs
Charles Bordenave; Djalil Chafai
Détail
Probability Surveys, 2012, 9, pp. 1-89
DOI
DOI : 10.1214/11-PS183
Début du résumé
These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting .....
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Scientific Book

Titre
Interactions between compressed sensing, random matrices, and high dimensional geometry
Auteurs
Djalil Chafai ; Olivier Guédon; Guillaume Lecué ; Alain Pajor
Détail
Société Mathématique de France, In press (Panoramas et Synthèses n°37), Dec. 2012
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Titre
Spectrum of Markov generators on sparse random graphs
Auteurs
Charles Bordenave; Pietro Caputo; Djalil Chafai
Détail
Feb. 2012. Accepted in Communications on Pure and Applied Mathematics (CPAM, 2013)
Début du résumé
We investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if k<>j and L(j,j)=-sum(L(j,k),k<>j), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős-Rényi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting .....
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2011

Article in peer-reviewed journal

Titre
Spectrum of non-Hermitian heavy tailed random matrices
Auteurs
Charles Bordenave; Pietro Caputo; Djalil Chafai
Détail
Communications in Mathematical Physics, 2011, 307 (2), pp. 513-560
DOI
DOI : 10.1007/s00220-011-1331-9
Début du résumé
Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}| is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X_{jk}, we prove that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability measure mu_alpha on C depending only on alpha such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1} (X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our approach combines Aldous & .....
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Titre
Spectrum of large random reversible Markov chains: heavy tailed weights on the complete graph
Auteurs
Charles Bordenave; Pietro Caputo; Djalil Chafai
Détail
Annals of Probability, 2011, 39 (4), pp. 1544-1590
DOI
DOI : 10.1214/10-AOP587
Début du résumé
We consider the random reversible Markov kernel K obtained by assigning i.i.d. non negative weights to the edges of the complete graph over n vertices, and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an alpha-stable law, with alpha in (0,2). When 1<= \alpha <2, we show that for a suitable regularly varying sequence kappa_n of index 1-1/alpha, the limiting spectral distribution mu_alpha of kappa_n K coincides with the one of the random symmetric matrix of the un-normalized weights (Levy matrix with i.i.d. entries). In contrast, when 0< alpha <1, .....
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Titre
Intertwining and commutation relations for birth-death processes
Auteurs
Djalil Chafai; Aldéric Joulin
Détail
Aug. 2011. Minor modifications. Accepted for publication in Bernoulli.
Début du résumé
Given a birth-death process on N with semigroup (P_t) and a discrete gradient d_u depending on a positive weight u, we establish intertwining relations of the form d_u P_t = Q_t d_u, where (Q_t) is the Feynman-Kac semigroup with potential V_u of another birth-death process. We provide applications when V_u is positive and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity. .....
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Titre
Central limit theorems for additive functionals of ergodic Markov diffusions processes
Auteurs
Patrick Cattiaux; Djalil Chafai; Arnaud Guillin
Détail
Apr. 2011
Début du résumé
We revisit functional central limit theorems for additive functionals of ergodic Markov diffusion processes. Translated in the language of partial differential equations of evolution, they appear as diffusion limits in the asymptotic analysis of Fokker-Planck type equations. We focus on the square integrable framework, and we provide tractable conditions on the infinitesimal generator, including degenerate or anomalously slow diffusions. We take advantage on recent developments in the study of the trend to the equilibrium of ergodic diffusions. We discuss examples and formulate open problems. .....
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2010

Article in peer-reviewed journal

Titre
Circular law for non-central random matrices
Auteurs
Djalil Chafai
Détail
Journal of Theoretical Probability, 2010, 23 (4), pp. 945-950
DOI
DOI : 10.1007/s10959-010-0285-8
Début du résumé
Let $(X_{jk})_{j,k\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean $0$ and variance $1$. Let $\la_{n,1},\ldots,\la_{n,n}$ be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. The strong circular law theorem states that with probability one, the empirical spectral distribution $\frac{1}{n}(\de_{\la_{n,1}}+\cdots+\de_{\la_{n,n}})$ converges weakly as $n\to\infty$ to the uniform law over the unit disc $\{z\in\dC;|z|\leq1\}$. In this short note, we provide an elementary argument that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\leq j,k\leq n}$ provided that $\mathrm{Tr}(MM^*)=O(n^2)$ and $\mathrm{rank}(M)=O(n^\al)$ with $\al<1$. Conveniently, the argument is similar to the one used for the non-central version of Wigner's and Marchenko-Pastur .....
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Titre
On the long time behavior of the TCP window size process
Auteurs
Djalil Chafai; Florent Malrieu; Katy Paroux
Détail
Stochastic Processes and their Applications, 2010, 120 (8), pp. 1518-1534
DOI
DOI : 10.1016/j.spa.2010.03.019
Début du résumé
The TCP window size process appears in the modeling of the famous Transmission Control Protocol used for data transmission over the Internet. This continuous time Markov process takes its values in $0,\infty)$, is ergodic and irreversible. It belongs to the Additive Increase Multiplicative Decrease class of processes. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. Several aspects of this process have already been investigated in the literature. In the present paper, we mainly get quantitative estimates for the convergence to equilibrium, in terms of the $W_1$ Wasserstein coupling distance, .....
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Titre
Asymptotic analysis and diffusion limit of the Persistent Turning Walker Model
Auteurs
Patrick Cattiaux; Djalil Chafai; Sébastien Motsch
Détail
Asymptotic Analysis, 2010, 67 (1-2), pp. 17-31
DOI
DOI : 10.3233/ASY-2009-0969
Début du résumé
The Persistent Turning Walker Model (PTWM) was introduced by Gautrais et al in Mathematical Biology for the modelling of fish motion. It involves a nonlinear pathwise functional of a non-elliptic hypo-elliptic diffusion. This diffusion solves a kinetic Fokker-Planck equation based on an Ornstein-Uhlenbeck Gaussian process. The long time diffusive'' behavior of this model was recently studied by Degond & Motsch using partial differential equations techniques. This model is however intrinsically probabilistic. In the present paper, we show how the long time diffusive behavior of this model can be essentially recovered and extended by using appropriate tools from stochastic analysis. The .....
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Titre
Spectrum of large random reversible Markov chains: two examples
Auteurs
Charles Bordenave; Pietro Caputo; Djalil Chafai
Détail
ALEA : Latin American Journal of Probability and Mathematical Statistics, 2010, 7, pp. 41-64
Début du résumé
We take on a Random Matrix theory viewpoint to study the spectrum of certain reversible Markov chains in random environment. As the number of states tends to infinity, we consider the global behavior of the spectrum, and the local behavior at the edge, including the so called spectral gap. Results are obtained for two simple models with distinct limiting features. The first model is built on the complete graph while the second is a birth-and-death dynamics. Both models give rise to random matrices with non independent entries. .....
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Titre
On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities
Auteurs
Djalil Chafai; Florent Malrieu
Détail
Annales de l'IHP - Probabilités et Statistiques, 2010, 46 (1), pp. 72-96
DOI
DOI : 10.1214/08-AIHP309
Début du résumé
Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals natural relations with some kind of band isoperimetry and support constrained .....
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Titre
The Dirichlet Markov Ensemble
Auteurs
Djalil Chafai
Détail
Journal of Multivariate Analysis, 2010, 101, pp. 555-567
DOI
DOI : 10.1016/j.jmva.2009.10.013
Début du résumé
We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d.\ rows following the Dirichlet distribution of mean $(1/n,\ldots,1/n)$. We show that if $\bM$ is such a random matrix, then the empirical distribution built from the singular values of$\sqrt{n}\,\bM$ tends as $n\to\infty$ to a Wigner quarter--circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of $\sqrt{n}\,\bM$ .....
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Other

Titre
Processus stochastiques en temps long
Auteurs
Djalil Chafai
Détail
[Conference digest]. Journées MAS et Journée en l'honneur de Jacques Neveu, Aug 2010, Talence, France
Début du résumé
Les modélisations de phénomènes d'évolution aléatoire issus de la biologie, de l'informatique, et de la physique constituent un thème riche et incontournable des mathématiques appliquées actuelles. Ces modélisations mettent en oeuvre des processus stochastiques, dont l'étude en temps long est fondamentale. Cette session parallèle est l'occasion de découvrir quelques aspects de ce vaste thème, comme la stabilité de diffusions inhomogènes, l'approximation de mesures quasi-stationnaires, ainsi que la stabilité de partages de fichiers en réseau. Cette session donne volontairement la parole à trois jeunes chercheurs, ainsi qu'à un spécialiste plus expérimenté. .....
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2009

Article in peer-reviewed journal

Titre
Confidence regions for the multinomial parameter with small sample size
Auteurs
Djalil Chafai; Didier Concordet
Détail
Journal of the American Statistical Association, 2009, 104 (487), pp. 1071-1079
DOI
DOI : 10.1198/jasa.2009.tm08152
Début du résumé
Consider the observation of n iid realizations of an experiment with d>1 possible outcomes, which corresponds to a single observation of a multinomial distribution M(n,p) where p is an unknown discrete distribution on {1,...,d}. In many applications, the construction of a confidence region for p when n is small is crucial. This concrete challenging problem has a long history. It is well known that the confidence regions built from asymptotic statistics do not have good coverage when n is small. On the other hand, most available methods providing non-asymptotic regions with controlled coverage are limited to the binomial case d=2. .....
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Titre
Aspects of large random Markov kernels
Auteurs
Djalil Chafai
Détail
Stochastics: An International Journal of Probability and Stochastic Processes, 2009, 81 (3-4), pp. 415-429
DOI
DOI : 10.1080/17442500903080314
Début du résumé
We briefly review certain asymptotic properties of random Markov kernels on a finite state space. These models can be thought of as finite Markov chains in random environment. Here, the asymptotics are taken with respect to the cardinality of the state space. We study, for instance, the behaviour of the normalized invariant vector, the global behaviour of the spectrum and the extremal eigenvalues. The analysis of these models involves Random Matrix Theory, convex compact polytopes and finite graphs with random weights. We also give some open problems related to these models. .....
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Titre
A new method for the estimation of variance matrix with prescribed zeros in nonlinear mixed effects models
Auteurs
Djalil Chafai; Didier Concordet
Détail
Statistics and Computing, 2009, 19 (2), pp. 129-138
DOI
DOI : 10.1007/s11222-008-9076-9
Début du résumé
We propose a new method for the Maximum Likelihood Estimator (MLE) of nonlinear mixed effects models when the variance matrix of Gaussian random effects has a prescribed pattern of zeros (PPZ). The method consists in coupling the recently developed Iterative Conditional Fitting (ICF) algorithm with the Expectation Maximization (EM) algorithm. It provides positive definite estimates for any sample size, and does not rely on any structural assumption on the PPZ. It can be easily adapted to many versions of EM. .....
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Titre
Comparison of nonparametric methods in nonlinear mixed effects models
Auteurs
Julie Antic; Céline Laffont; Djalil Chafai; Didier Concordet
Détail
Computational Statistics & Data Analysis / Computational Statistics and Data Analysis, 2009, 53, pp. 642-656
DOI
DOI : 10.1016/j.csda.2008.08.021
Début du résumé
During the drug development, nonlinear mixed effects models are routinely used to study the drug's pharmacokinetics and pharmacodynamics. The distribution of random effects is of special interest because it allows to describe the heterogeneity of the drug's kinetics or dynamics in the population of individuals studied. Parametric models are widely used, but they rely on a normality assumption which may be too restrictive. In practice, this assumption is often checked using the empirical distribution of random effects' empirical Bayes estimates. Unfortunately, when data are sparse (like in patients phase III clinical trials), this method is unreliable. In this context, nonparametric .....
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2008

Article in peer-reviewed journal

Titre
On gradient bounds for the heat kernel on the Heisenberg group
Auteurs
Dominique Bakry; Fabrice Baudoin; Michel Bonnefont; Djalil Chafai
Détail
Journal of Functional Analysis, 2008, 255 (8), pp. 1905-1938
DOI
DOI : 10.1016/j.jfa.2008.09.002
Début du résumé
It is known that the couple formed by the two dimensional Brownian motion and its Lévy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov .....
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Titre
Explicit formulas for a continuous stochastic maturation model. Application to anticancer drug pharmacokinetics/pharmacodynamics.
Auteurs
Djalil Chafai; Didier Concordet
Détail
Stochastic Models, 2008, 24 (3), pp. 376-400
DOI
DOI : 10.1080/15326340802232244
Début du résumé
We present a continuous time model of maturation and survival, obtained as the limit of a compartmental evolution model when the number of compartments tends to infinity. We establish in particular an explicit formula for the law of the system output under inhomogeneous killing and when the input follows a time-inhomogeneous Poisson process. This approach allows the discussion of identifiability issues which are of difficult access for finite compartmental models. The article ends up with an example of application for anticancer drug pharmacokinetics/pharmacodynamics. .....
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Habilitation à diriger des recherches

Titre
Contributions à l'étude de modèles biologiques, d'inégalités fonctionnelles, et de matrices aléatoires
Auteurs
Djalil Chafai
Détail
Université Paul Sabatier - Toulouse III, Oct. 2008. French
Début du résumé
Les travaux présentés concernent trois thématiques autonomes :

(1) Modèles biologiques et statistique : modèles compartimentaux, pharmacocinétique et pharmacodynamie de population, estimateurs pour problèmes inverses stochastiques, modèles non-linéaires à effets mixtes, modèles de mélanges, algorithmes de type EM et ICF, modèles graphiques de covariance, modélisation en cancérologie, processus ponctuels, particules, files d'attentes, renormalisation de processus markoviens inhomogènes et formules de Feynman-Kac

(2) Inégalités fonctionnelles : inégalités de type Sobolev, concentration de la mesure, isopérimétrie rôle de la convexité dans les inégalités entropiques, tensorisation, noyau de la chaleur, groupe d'Heisenberg et dynamiques hypoelliptiques, files d'attentes, mélanges de lois
.....
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2007

Article in peer-reviewed journal

Titre
On the strong consistency of asymptotic M-estimators
Auteurs
Djalil Chafai; Didier Concordet
Détail
Journal of Statistical Planning and Inference, 2007, 137 (9), pp. 2774-2783
DOI
DOI : 10.1016/j.jspi.2006.09.027
Début du résumé
The aim of this article is to simplify Pfanzagl's proof of consistency for asymptotic maximum likelihood estimators, and to extend it to more general asymptotic M-estimators. The method relies on the existence of a sort of contraction of the parameter space which admits the true parameter as a fixed point. The proofs are short and elementary. .....
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Scientific Book

Titre
Modélisation stochastique et simulation - Cours et applications
Auteurs
Bernard Bercu ; Djalil Chafai
Détail
SMAI. Dunod, pp. 352, Oct. 2007, Mathématiques appliquées pour le Master - Sciences Sup
Début du résumé
Ce livre place la simulation au coeur des probabilités et des statistiques. Il est principalement destiné aux étudiants qui ont déjà suivi un enseignement de base dans ces domaines. L'accent est volontairement mis sur la structure et sur l'intuition. La rédaction associe résultats théoriques, modèles et algorithmes stochastiques, ainsi qu'une variété d'applications illustrées par des programmes informatiques. L'ouvrage est destiné aux étudiants en Master de mathématiques appliquées, aux élèves ingénieurs, aux candidats au Capes et à l'Agrégation, aux doctorants, ainsi qu'aux curieux explorateurs de l'aléatoire. .....
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2006

Article in peer-reviewed journal

Titre
Binomial-Poisson entropic inequalities and the M/M/$\infty$ queue
Auteurs
Djalil Chafai
Détail
ESAIM P&S, 2006, 10, pp. 317-339
Début du résumé
This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/$\infty$ queue. They describe in particular the exponential dissipation of $\Phi$-entropies along this process. This simple queueing process appears as a model of constant curvature'', and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/$\infty$ queues. Proofs are elementary .....
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Titre
On nonparametric maximum likelihood for a class of stochastic inverse problems
Auteurs
Djalil Chafai; Jean-Michel Loubes
Détail
Statistics and Probability Letters, 2006, 76 (12), pp. 1225-1237
DOI
DOI : 10.1016/j.spl.2005.12.019
Début du résumé
We establish the consistency of a nonparametric maximum likelihood estimator for a class of stochastic inverse problems. We proceed by embedding the framework into the general settings of early results of Pfanzagl related to mixtures. .....
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2005

Preprint, Working Paper, ...

Titre
Inégalités de Poincaré et de Gross pour les mesures de Bernoulli, de Poisson, et de Gauss
Auteurs
Djalil Chafai
Détail
Oct. 2005
Début du résumé
Les inégalités de Sobolev logarithmiques doivent leur nom à un article célèbre de Gross paru en 1975. Ces inégalités fonctionnelles apparaissent en particulier comme une expression de la propriété d'hypercontractivité de semi-groupes markoviens, comme une traduction de la décroissance exponentielle de l'entropie le long de ces semi-groupes, comme une information de concentration gaussienne de la mesure, et enfin comme une façon de renforcer les inégalités de Poincaré, plus traditionnelles en analyse. Dans cet article, des inégalités fonctionnelles optimales de ce type sont obtenues pour les mesures de Poisson et de Gauss, par tensorisation infinie, à partir d'inégalités optimales pour des .....
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2004

Article in peer-reviewed journal

Titre
Entropies, convexity, and functional inequalities : On Phi-entropies and Phi-Sobolev inequalities
Auteurs
Djalil Chafai
Détail
Journal of Mathematics of Kyoto University, 2004, 44 (2), pp. 325-363
Début du résumé
Our aim is to provide a short and self contained synthesis which generalise and unify various related and unrelated works involving what we call Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies can be seen in particular as an inclusive interpolation between Poincare and Gross logarithmic Sobolev inequalities. In addition to the known material, extensions are provided and improvements are given for some aspects. Stability by tensor products, convolution, and bounded perturbations are addressed. We show that under simple convexity assumptions on Phi, such inequalities hold in a lot of situations, including hyper-contractive diffusions, uniformly strictly log-concave measures, Wiener measure .....
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2003

Article in peer-reviewed journal

Titre
Gaussian maximum of entropy and reversed log-Sobolev inequality
Auteurs
Djalil Chafai
Détail
Séminaire de Probabilités XXXVI, 2003, Lecture Notes in Mathematics 1801, pp. 194-200
DOI
DOI : 10.1007/b10068
Début du résumé
The aim of this note is to connect a reversed form of the Gross logarithmic Sobolev inequality with the Gaussian maximum of Shannon's entropy power. There is thus a complete parallel with the well-known link between logarithmic Sobolev inequalities and their information theoretic counterparts. We moreover provide an elementary proof of the reversed Gross inequality via a two-point inequality and the Central Limit Theorem. .....
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Titre
Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems
Auteurs
Djalil Chafai
Détail
Markov Processes and Related Fields, 2003, 9 (3), pp. 341-362
Début du résumé
Inspired by the recent results of C. Landim, G. Panizo and H.-T. Yau LPY on spectral gap and logarithmic Sobolev inequalities for unbounded conservative spin systems, we study uniform bounds in these inequalities for Glauber dynamics of Hamiltonian of the form V(x_1) + ... + V(x_n) + V(M-x_1 -...-x_n), (x_1,...,x_n) in R^n Specifically, we examine the case V is strictly convex (or small perturbation of strictly convex) and, following LPY, the case V is a bounded perturbation of a quadratic potential. By a simple path counting argument for the standard random walk, uniform bounds for the Glauber dynamics yields, in .....
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2002

Article in peer-reviewed journal

Titre
Channel selection methods for infrared atmospheric sounding interferometer radiances
Auteurs
Florence Rabier; Nadia Fourrié; Djalil Chafai; Pascal Prunet
Détail
Quarterly Quarterly Journal of the Royal Meteorological Society, 2002, 128 (581), pp. 1011-1027
DOI
DOI : 10.1256/0035900021643638
Début du résumé
Advanced infrared sounders will provide thousands of radiance data at every observation location. The number of individual pieces of information is not usable in an operational numerical weather-prediction context, and we have investigated the possibilities of choosing an optimal subset of data. These issues have been addressed in the context of optimal linear estimation theory, using simulated Infrared Atmospheric Sounding Interferometer data. Several methods have been tried to select a set of the most useful channels for each individual atmospheric profile. These are two methods based on the data resolution matrix, one method based on the Jacobian matrix, and one .....
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PhD thesis

Titre
Sur les inégalités de Sobolev logarithmiques en théorie de l'information et pour des systèmes de spins conservatifs en mécanique statistique
Auteurs
Djalil Chafai
Détail
Université Paul Sabatier - Toulouse III, May. 2002. French
Début du résumé
1°) Utilisation d'inégalités fonctionnelles de Bobkov pour l'établissement de principes de grandes déviations quasi-gaussiens.

2°) Etude de l'inégalité de Sobolev logarithmique en théorie de l'information.

3°) Etablissement d'inégalités de Poincaré et de Sobolev logarithmiques pour certaines dynamiques de Kawasaki et Glauber pour un modèle à spins continus en mécanique statistique. .....
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2000

Scientific Book

Titre
Sur les inégalités de Sobolev logarithmiques
Auteurs
Cécile Ané; Sébastien Blachère; Djalil Chafai; Pierre Fougères; Ivan Gentil; Florent Malrieu; Cyril Roberto; Grégory Scheffer
Détail
Panoramas et Synthèses. Société Mathématique de France, pp. xvi+217, 2000
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1999

Article in peer-reviewed journal

Titre
Méthodes fonctionnelles pour des grandes déviations quasi-gaussiennes
Auteurs
Djalil Chafai; Michel Ledoux
Détail
Comptes Rendus de l Académie des Sciences - Series I - Mathematics, 1999, 329 (6), pp. 523-526
DOI
DOI : 10.1016/S0764-4442(00)80054-3
Début du résumé
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. .....
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