Abstract
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.
Citation
Alexander E. Litvak. Konstantin Tikhomirov. "Order statistics of vectors with dependent coordinates, and the Karhunen–Loève basis." Ann. Appl. Probab. 28 (4) 2083 - 2104, August 2018. https://doi.org/10.1214/17-AAP1321
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