-
Saarland University,
Faculty of Mathematics,
Room 2.15, Building E2 4.Email: banna@math.uni-sb.de
CV: [anglais]
Thèse de doctorat:[PDF]
ResearchGate:
Google Scholar:
LinkedIn:
Je suis à New York University Abu Dhabi depuis Janvier 2021.
Je suis actuellement en post-doctorat à l'Université de la Sarre dans l'équipe de probabilités libres avec Roland Speicher. Précédemment, j'ai effectué un post-doctorat à Télécom ParisTech sous l'encadrement de Walid Hachem. Ma thèse de doctorat a été effectuée sous la direction de Florence Merlevède et Emmanuel Rio à l'Université Paris-Est Marne-la-Vallée (UPEM), au sein du Laboratoire d'Analyse et de Mathématiques Appliquées.
Domaine de recherche:- Matrices aléatoires
- Inégalités de concentration
- Dépendance faible
- Probabilités libres
- Berry-Esseen bounds for the multivariate B-free CLT and operator-valued matrices.
En collaboration avec T. Mai.
arXiv - Résumé.Résumé. We provide bounds of Berry-Esseen type for fundamental limit theorems in operator-valued free probability theory such as the operator-valued free Central Limit Theorem and the asymptotic behaviour of distributions of operator-valued matrices. Our estimates are on the level of operator-valued Cauchy transforms and the Lévy distance. We address the single-variable as well as the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the Lévy distance for the operator-valued free Central Limit Theorem. Our results also yield quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile. In the scalar-valued multivariate case, these estimates could be passed to explicit bounds on the order of convergence under the Kolmogorov distance.- Matrix Poincaré Inequalities and Concentration.
En collaboration avec R. Aoun et P. Youssef.
Advances in Mathematics.
Elsevier - arXiv - Résumé.Résumé. We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the noncommutative setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincaré inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong Rayleigh measures. The latter represents the first instance of matrix concentration for general matrix functions of negatively dependent random variables.- Operator-valued matrices with free or exchangeable entries.
En collaboration avec G. Cébron.
arXiv - Résumé.Résumé . We study matrices whose entries are free or exchangeable noncommutative elements in some tracial $W^*$-probability space. More precisely, we consider operator-valued Wigner and Wishart matrices and give for the associated Cauchy transforms explicit rates of convergence to operator-valued semicircular elements over some subalgebra. Applications to block random matrices with a Wigner or circulant structure and in independent or correlated blocks are also given. Our approach relies on a noncommutative extension of the Lindeberg method and Gaussian interpolation techniques.- Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information.
En collaboration avec T. Mai.
Journal of Functional Analysis.
Elsevier - arXiv - Abstract.ABSTRACT. This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information. It is shown that their analytic distributions have Hölder continuous cumulative distribution functions with an explicit Hölder exponent that depends only on the degree of the considered polynomial. This, in particular, guarantees that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy. We further provide a general criterion that gives for weak approximations of measures having Hölder continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance. Finally, we apply these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general Gibbs laws.- A CLT for linear spectral statistics of large random information-plus-noise matrices.
En collaboration avec J. Najim et J. Yao.
Stochastic Processes and their Applications.
Elsevier - HAL - Résumé.Résumé . Consider a matrix\[\mathrm{Y}_n= \frac{\sigma}{\sqrt{n}} \mathrm{X}_n +\mathrm{A}_n, \] where $\sigma>0 $ and $\mathrm{X}_n$ is an $N\times n$ random matrix with i.i.d. real or complex standardized entries and $\mathrm{A}_n$ is a $N\times n$ deterministic matrix with bounded spectral norm. The fluctuations of the linear spectral statistics of the eigenvalues: \[\mathrm{Trace}\, f(\mathrm{Y}_n \mathrm{Y}_n^*)= \sum_{i=1}^N f(\lambda_i),\quad (\lambda_i)\ \mathrm{eigenvalues\ of}\ \mathrm{Y}_n \mathrm{Y}_n^*\] are shown to be gaussian, in the case where $f$ is a smooth function of class $C^3$ with bounded support, and in the regime where both dimensions of matrix $\mathrm{Y}_n$ go to infinity at the same pace. The CLT is expressed in terms of vanishing Lévy-Prohorov distance between the linear statistics' distribution and a centered Gaussian probability distribution, the variance of which depends upon $N$ and $n$ and may not converge. The proof combines ideas from Bai and Silverstein (2004), Hachem et al. (2012) and Najim and Yao (2013). - Bernstein type inequality for a class of dependent random matrices
Random Matrices: Theory and Applications.
En collaboration avec F. Merlevède et P. Youssef
World Scientific - arXiv - Résumé.Résumé . In this paper we obtain a Bernstein type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. (2009) in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices. - Limiting spectral distribution of Gram matrices associated with functionals of β-mixing processes
Journal of Mathematical Analysis and Applications.
Elsevier - arXiv - Résumé.Résumé. We give asymptotic spectral results for Gram matrices whose entries are dependent across both rows and columns. More precisely, they consist of short or long range dependent random variables having moments of second order and that are functionals of an absolutely regular sequence. We also give a concentration inequality of the Stieltjes transform and we prove that, under an arithmetical decay condition on the β-mixing coefficients, it is almost surely concentrated around its expectation. Applications to examples of positive recurrent Markov chains and dynamical systems are also given. - On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries
Stochastic Processes and their Applications.
En collaboration avec F. Merlevède et M. Peligrad
Elsevier - PDF - Résumé.Résumé . For symmetric random matrices with correlated entries, which are functions of independent random variables, we show that the asymptotic behavior of the empirical eigenvalue distribution can be obtained by analyzing a Gaussian matrix with the same covariance structure. This class contains both cases of short and long range dependent random fields. The technique is based on a blend of blocking procedure and Lindeberg's method. This method leads to a variety of interesting asymptotic results for matrices with dependent entries, including applications to linear processes as well as nonlinear Volterra-type processes entries. - Limiting spectral distribution of large sample covariance matrices associated with a class of stationary processes
Journal of Theoratical Probability.
En collaboration avec F. Merlevède
Springer - PDF - Résumé.Résumé. In this paper we derive an extension of the Marcenko-Pastur theorem to a large class of weak dependent sequences of real-valued random variables having only moment of order 2. Under a mild dependence condition that is easily verifiable in many situations, we derive that the limiting spectral distribution of the associated sample covariance matrix is characterised by an explicit equation for its Stieltjes transform, depending on the spectral density of the underlying process. Applications to linear processes, functions of linear processes and ARCH models are given.
- Gaussian Hilbert Spaces together avec Tobias Mai.
- Random matrices avec Roland Speicher (2018).
- Master 2 Recherche Labex Bézout, bourse délivrée par l'Université Paris-Est (Interview).
- Prix d'excellence délivré par l’Université Libanaise.
Activités Scientifiques:
- Septembre 2014- Septembre 2015: Organisatrice du Séminaire des Doctorants du LAMA, UPEM-UPEC, France.
Séjours à l'étranger:
- 26 - 29 Août 2019: American University of Beirut: invitée par Richard Aoun.
- 15 - 19 Juillet 2019: Université Paris Diderot: invitée par Pierre Youssef.
- 12 - 16 Novembre 2018: Université Paul Sabatier: invitée par Guillaume Cébron.
- 8 - 12 Mai 2017: Université Paul Sabatier: invitée par Guillaume Cébron.
- 28 Septembre - 16 Octobre 2014: University of Alberta: invitée par Nicole Tomczak-Jaegermann et Alexander Litvak pour une durée de trois semianes.
- 8 - 12 Juin 2020: Free Probability - on the occasion of Roland Speicher's 60th birthday, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany (Reporté).
- 17 - 21 Août 2020: International Workshop on Operator Theory and its Applications IWOTA 2020, Lancaster University, Lancaster, UK (Reporté).
- 24 - 28 Août 2020: 15th French-Romanian Conference on Applied Mathematics, Transilvania University of Brasov, Brasov, Romania (Reporté).
- 5 - 9 Avril 2021: Phenomena in High Dimensions, CIRM, Marseille, France.
- 25 Juillet 2019: 30th International Workshop on Operator Theory and its Applications, Instituto Superior Técnico, University of Lisbon, Portugal.
- 20 Juin 2019: Journées IOPS - Image, Optimisation, Probabilités et Statistique, Réserve Ornithologique du Teich, Le Teich, France.
- 14 Mai 2019: Séminaire d'Analyse Harmonique Non Commutative, Université de Caen, Caen, France.
- 2 Mai 2019: Random Matrix Theory: Applications in the Information Era, Jagellonian University, Cracovie, Polgne.
- 6 Mars 2019: Atelier Probabilités libres: la théorie, ses extensions pendant un programme thématique d'un mois sur les Nouveaux Développements en Probabilités Libres et Applications, CRM, Montréal, Canada.
- 5 Décembre 2018: Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Allemagne.
- 17 Novembre 2017: Distributional symmetries and independences, Bordeaux, France.
- 8 Novembre 2017: Seminario Interinstitucional de Matrices Aleatorias: Probabilidad Libre, CIMAT, Guanajuato, Mexique.
- 9 Février 2017: Probability Seminar, Université du Luxembourg, Luxembourg.
- 9 Décembre 2016: A Bernstein inequality for dependent random matrices. Analytic versus Combinatorial in Free Probability, Banff International Research Station for Mathematical Innovation and Discovery, Banff, Canada. [video], [PDF].
- 6 Octobre 2016: Séminaire modélisation stochastique, Université Paris-Diderot, Paris, France.
- 6 Septembre 2016: Universität des Saarlandes
- 31 Août 2016: Journées MAS 2016, Grenoble, France.
- 7 Juin 2016: Séminaire de Probabilités-Statistiques, Université de Versailles, France.
- 30 Mai 2016: Groupe de travail Probabilités, Théorie Ergodique et Systèmes Dynamiques, Rouen, France.
- 17 Mars 2016: Rencontre: Martingales, chaînes de Markov et Systèmes dynamiques, Landeda, France.
- 3 Décembre 2015: Journées du Gdr Analyse Fonctionnelle, Harmonique et Probabilités, CIRM, France.
- 27 Novembre 2015: Séminaire Calcul stochastique, Université de Strasbourg, France.
- 26 Mai 2015: Journées de probabilités 2015, Institut de Mathématique de Toulouse, France.
- 16 Janvier 2015: Groupe de travail Matrices et graphes aléatoires (Mega) IHP, France.
- 7 Octobre 2014: University of Alberta, Edmonton, Canada.
- 1 Octobre 2014: University of Alberta, Edmonton, Canada.
- 26 Mai 2014:Young Women in Probability, Bonn, Allemagne.
- 20 Mai 2014: Séminaire de Probabilités-Statistiques, Université de Versailles, France.
- 30 Avril 2014: Séminaire des Doctorants, UPEC, France.
- 7 Avril 2014: Onzième Colloque Jeunes Probabilistes et Statisticiens, Forges-les-Eaux, France.
- 19 Février 2013: Groupe de travail ASPRO, UPEM, France.
- Mars 2015- Septembre 2015: Représentante des Doctorants au Conseil du laboratoire.
Marwa Banna - Université de la Sarre - Allemagne - Bernstein type inequality for a class of dependent random matrices
- A CLT for linear spectral statistics of large random information-plus-noise matrices.
- Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information.
- Operator-valued matrices with free or exchangeable entries.
- Matrix Poincaré Inequalities and Concentration.