Order statistics of vectors with dependent coordinates, and the Karhunen.Loève basis

Let X be an n-dimensional random centered Gaussian vector with independent but not identically distributed coordinates and let T be an orthogonal transformation of Rn. We show that the random vector Y=T(X) satisfies Ek.j=1j-mini.nXi2.CEk.j=1j-mini.nYi2 for all k.n, where .j-min. denotes the jth 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.Loève 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.