Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\R^n$. We show that the random vector $Y=T(X)$ satisfies
$$
\mathbb{E} \sum \limits_{j=1}^k j\mobx{-}\min _{i\leq n}{X_{i}}^2 \leq C \mathbb{E} \sum\limits_{j=1}^k j\mobx{-}\min _{i\leq n}{Y_{i}}^2
$$
for all $k\leq n$, where ``$\jm$'' 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--Lo\`eve basis for the nonlinear
reconstruction. We also show some relations for order statistics of random vectors
(not only Gaussian) which are of independent interest. This is a joint work with Konstantin Tikhomirov. Non UBC Unreviewed Author affiliation: University of Alberta Faculty
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Citation
APA Citation:
Litvak, A. (2017). Order statistics of vectors with dependent coordinates. [Data set]. UBC cIRcle BIRS Workshop Lecture Videos. http://hdl.handle.net/2429/63692