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Order statistics of vectors with dependent coordinates. Litvak, Alexander


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 ``$\jm$'' 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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