# Mathematical Sciences Research Institute

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# Seminar

GFA Main Seminar: Order statistics of vectors with dependent coordinates September 08, 2017 (11:30 AM PDT - 12:30 PM PDT)
Parent Program: Geometric Functional Analysis and Applications MSRI: Baker Board Room
Speaker(s) Alexander Litvak (University of Alberta)
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Abstract/Media

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

$$\mathbb{E} \sum \limits_{j=1}^k j\mobx{-}\min _{i\leq n}{X_{i}}^2 \leq C \mathbb{E} \sum\limits_{j=1}^k j\mobx{-}\min _{i\leq n}{Y_{i}}^2$$

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.

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