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

@article{Litvak2018OrderSO,
  title={Order statistics of vectors with dependent coordinates, and the Karhunen–Lo{\`e}ve basis},
  author={Alexander E. Litvak and Konstantin E. Tikhomirov},
  journal={The Annals of Applied Probability},
  year={2018}
}
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 $\mathbb R^n$. We show that the random vector $Y=T(X)$ satisfies $$\mathbb E\sum\limits_{j=1}^k j\mbox{-}\min_{i\leq n}{X_{i}}^2 \leq C\mathbb E\sum\limits_{j=1}^k j\mbox{-}\min_{i\leq n}{Y_{i}}^2$$ for all $k 0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni… Expand
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