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
Description
Le Jeudi à 10h30 - IMJ-PRG - 4 place Jussieu - 75005 PARIS
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Orateur(s)
Alexander Litvak - Edmonton,
Titre
Order statistics of vectors with dependent coordinates
Date
21/06/2018
Horaire
10:30 à 17:36
Diffusion
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
\mathbb
E
\sum \limits_
j=1
^k j\mbox
-
\min _
i\leq n
X_
i^2 \leq C \mathbb
E
\sum\limits_
j=1
^k j\mbox
-
\min _
i\leq n
Y_
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.
Salle
salle 13 - couloir 15-16 - 4ème étage
Adresse
Campus Pierre et Marie Curie
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