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SUMMARY:Order statistics and Mallat--Zeitouni problem - Alexander Litvak (
 University of Alberta)
DTSTART:20190218T142000Z
DTEND:20190218T145500Z
UID:TALK119953@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Let $X$ be an $n$-dimensional random centered Gaussian vector 
 with independent but not necessarily identically distributed coordinates a
 nd let $T$ be an orthogonal transformation of $mathbb{R}^n$. We show that 
 the random vector $Y=T(X)$ satisfies $$mathbb{E} sum limits_{j=1}^k jmbox{
 -}min _{ileq n}{X_{i}}^2 leq C mathbb{E} sumlimits_{j=1}^k jmbox{-}min _{i
 leq n}{Y_{i}}^2$$ for all $kleq n$\, where ``$jmbox{-}min$&#39\;&#39\; den
 otes the $j$-th smallest component of the corresponding vector and $C>0$ i
 s a universal constant. This resolves (up to a multiplicative constant) an
  old question of S.Mallat and O.Zeitouni regarding optimality of the Karhu
 nen--Lo`eve basis for the nonlinear reconstruction. We also show some rela
 tions for order statistics of random vectors (not only Gaussian)\, which a
 re of independent interest. This is a joint work with Konstantin Tikhomiro
 v.
LOCATION:Seminar Room 1\, Newton Institute
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