Modern methods of multivariate statistics - Couverture souple

9782705684198: Modern methods of multivariate statistics

Couverture souple

ISBN 10 : 2705684190 ISBN 13 : 9782705684198

Editeur : Hermann Glassin, 2014

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Les statistiques multivari�es forment une partie autonome des statistiques math�matiques et jouent un r�le important dans les applications, notamment dans l'�tude des caract�res gaussiens multidimensionnels et de leur matrice de covariance. Les chapitres de ce livre pr�sentent des aspects diff�rents de la grande vari�t� des m�thodes math�matiques qui sont actuellement utilis�es dans ce domaine des statistiques : analyse de Fourier, th�orie des graphes et des mod�les graphiques, alg�bre et analyse matricielle, analyse harmonique sur les groupes, analyse stochastique d'It�, matrices al�atoires, th�orie du potentiel.

Les informations fournies dans la section � Synopsis � peuvent faire r�f�rence � une autre �dition de ce titre.

Pr�sentation de l'�diteur

Modem methods of multivariate statistics

J. Faraut, P. Graczyk, H. Ishi, G. Letac, H. Massam & E. Mayerhofer
Editors : Piotr Graczyk & Abdelhamid Hassairi

Multivariate statistical analysis is an autonomous branch of mathematical statistics and it plays an important role in applications, especially in the study of multidimensional Gaussian statistics and of their covariance matrix.
The lectures of this book present different aspects and the great variety of mathematical tools and methods which are used today in this domain of statistics : Fourier analysis, graphs theory, graphical models theory, matrix analysis and algebra, harmonic analysis on groups, It� stochastic analysis, random matrices, potential theory.

These topics were exposed during a Workshop CIMPA-FECYT-UNESCO-ANR which was held in Hammamet, Tunisia, in September 2011, by J. Faraut, P. Graczyk, H. Ishi, G. Letac, H. Massam and E. Mayerhofer.

Les statistiques multivari�es forment une partie autonome des statistiques math�matiques et jouent un r�le important dans les applications, notamment dans l'�tude des caract�res gaussiens multidimensionnels et de leurs matrices de covariance. Les chapitres de ce livre pr�sentent diff�rents aspects de la grande vari�t� des m�thodes math�matiques qui sont actuellement utilis�es dans ce domaine des statistiques : analyse de Fourier, th�orie des graphes et des mod�les graphiques, alg�bre et analyse matricielle, analyse harmonique sur les groupes, analyse stochastique d'It�, matrices al�atoires, th�orie du potentiel.

Ces mati�res ont �t� expos�es par J. Faraut, P. Graczyk, H. Ishi, G. Letac, H. Massam et E. Mayerhofer lors d'une �cole CIMPA-FECYT-UNESCO-ANR qui a eu lieu � Hammamet en Tunisie, en septembre 2011.

Extrait

Logarithmic potential theory, orthogonal polynomials, and random matrices

Jacques Faraut

In this course we present the relations which exist between

- The logarithmic potential theory,
- The asymptotic distribution of the zeros of classical orthogonal polynomials,
- The asymptotic distribution of the eigenvalues of random matrices.

Stieltjes considers a finite System of n electric charges moving freely on a line and submitted to an external field. He observes that the minimum of the electrostatistic energy of the System is attained at the points whose coordinates are zeros of certain classical orthogonal polynomials of degree n. This is the electrostatic model for orthogonal polynomials.

In order to study the asymptotic distribution of the zeros of a classical orthogonal polynomial pn of degree n as n → ��, one uses a basic result in logarithmic potential theory about the energy and equilibrium measures.

This result is also used in random matrix theory. This is the log gas model of Dyson. In particular, using logarithmic potential theory, one obtains a proof of Wigner Theorem about the convergence of the statistical distribution of the eigenvalues to the semicircle law in case of the Gaussian Orthogonal Ensemble, and the Gaussian Unitary Ensemble.

We will also consider the question : What is the probability for a symmetric matrix to be positive definite ? We will present recent results by Dean and Majumdar whose proof also uses the logarithmic potential theory.

We thank the organizers of the Hammamet CIMPA school for inviting us to present this topic : Piotr Graczyk, Abdelhamid Hassairi, and G�rard Letac. We are very grateful to Marouane Rabaoui who realized the histograms and graphics which illustrate clearly the main results presented in these notes. We also thank the referee for his careful reading of the manuscript.

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