The Annals of Applied Probability

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

Alexander E. Litvak and Konstantin Tikhomirov

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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 $\mathbb{R}^{n}$. We show that the random vector $Y=T(X)$ satisfies \begin{equation*}\mathbb{E}\sum_{j=1}^{k}j\mbox{-}\min_{i\leq n}{X_{i}}^{2}\leq C\mathbb{E}\sum_{j=1}^{k}j\mbox{-}\min_{i\leq n}{Y_{i}}^{2}\end{equation*} for all $k\leq n$, where “$j\mbox{-}\min$” 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.

Article information

Ann. Appl. Probab., Volume 28, Number 4 (2018), 2083-2104.

Received: December 2016
Revised: May 2017
First available in Project Euclid: 9 August 2018

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Zentralblatt MATH identifier

Primary: 62G30: Order statistics; empirical distribution functions 60E15: Inequalities; stochastic orderings 60G15: Gaussian processes 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 94A08: Image processing (compression, reconstruction, etc.) [See also 68U10]

Order statistics Karhunen–Loève basis nonlinear approximation INID case


Litvak, Alexander E.; Tikhomirov, Konstantin. Order statistics of vectors with dependent coordinates, and the Karhunen–Loève basis. Ann. Appl. Probab. 28 (2018), no. 4, 2083--2104. doi:10.1214/17-AAP1321.

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