Séminaires : Séminaire d'Analyse Fonctionnelle

Equipe(s) : af,
Responsables :E. Abakoumov - A.Eskenazis - D. Cordero-Erausquin - M. Fathi - O. Guédon - B. Maurey
Email des responsables :
Salle : salle 13 - couloir 15-16 - 4ème étage
Adresse :Campus Pierre et Marie Curie
Le Jeudi à 10h30 -  IMJ-PRG - 4 place Jussieu - 75005 PARIS

Orateur(s) Alexander Litvak - Edmonton,
Titre Order statistics of vectors with dependent coordinates
Horaire10:30 à 17:36
Résume Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\R^n$. We show that the random vector $Y=T(X)$ satisfies $\displaystyle
\mathbbE \sum \limits_j=1^k j\mbox-\min _i\leq nX_i^2 \leq C \mathbbE \sum\limits_j=1^k j\mbox-\min _i\leq nY_i^2
$ for all $k\leq n$, where ``$ j\mobx-\min$'' denotes the $j$-th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the Karhunen--Lo\`eve basis for the nonlinear reconstruction. We also show some relations for order statistics of random vectors (not only Gaussian), which are of independent interest. This is a joint work with Konstantin Tikhomirov.
Sallesalle 13 - couloir 15-16 - 4ème étage
AdresseCampus Pierre et Marie Curie