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 E k ∑ j=1 jmin i≤n Xi 2 ≤ CE k ∑ j=1 jmin i≤n Yi 2 for all k ≤ n, where “jmin” 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–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. AMS 2010 Classification: 62G30, 60E15, 60G15, 60G35, 94A08