  Let $X$ be an $n$-dimensional random centered Gaussian vector with
independent but not identically distributed coordinates and let $T$ be an
orthogonal trasformation 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 $k<n$, where "$j\mbox{-}\min$" denotes the $j$-th
smallest component of 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-Loeve basis for the nonlinear
signal approximation. As a by-product we obtain some relations for order
statistics of random vectors (not only Gaussian) which are of independent
interest.
