Abstract

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 $$mathbb{E} sum limits_{j=1}}k jmbox{}min {X{i}}^{2 leq C mathbb{E} sumlimits_{j=1}}k jmbox{}min {Y{i}}^2$$ for all $kleq n$, where ``$jmbox{}min$'' 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.