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

2016/09/07 by Alexander E. Litvak, Konstantin Tikhomirov, Litvak, Alexander E. +1 · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1609.02126

minor fixes

arxiv created 2017/05/28 · arxiv updated 2017/05/30

Abstract

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 Rⁿ. We show that the random vector Y=T(X) satisfies \mathbb E∑_j=1^k j-min_{i≤ n}X_i² ≤ C\mathbb E∑_j=1^k j-min_{i≤ n}Y_i² for all k<n, where "j-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.

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