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Order statistics and Mallat--Zeitouni problem

Presented by: 
Alexander Litvak
Monday 18th February 2019 - 14:20 to 14:55
INI Seminar Room 1
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 $\mathbb{R}^n$. We show that the random vector $Y=T(X)$ satisfies $$\mathbb{E} \sum \limits_{j=1}^k j\mbox{-}\min _{i\leq n}{X_{i}}^2 \leq C \mathbb{E} \sum\limits_{j=1}^k j\mbox{-}\min _{i\leq n}{Y_{i}}^2$$ for all $k\leq n$, where ``$j\mbox{-}\min$'' 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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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons