ORDER STATISTICS OF VECTORS WITH DEPENDENT COORDINATES, AND THE KARHUNEN-LOEVE BASIS

• 成果类型: Article
• 署名作者: Litvak, Alexander E.; Tikhomirov, Konstantin
• 署名单位: University of Alberta; University of Alberta; Princeton University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1321
• 发表日期: 2018
• 页码: 2083-2104
• 关键词: sequences; NORMS

摘要

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 R-n. We show that the random vector Y = T (X) satisfies E Sigma j=1(k) j-min (i)(<= n) X-i(2 )<= CE Sigma j=1(k) j-min(i)(<= n) Y-i(2 ) for all k <= n, where j-min denotes the jth 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.