Let $X=(X_1,\dots,X_n)$ and $Y=(Y_1,\dots,Y_n)$ be centered Gaussian vectors with variance matrix $\Gamma_X$ and $\Gamma_Y$. We assume that the matrix $\Gamma_Y-\Gamma_X$ is positive definite. Is it possible to prove that $$ \forall x \ge 0, \mathbb{P}(\max \lbrace |Y_1|,\dots,|Y_n| \rbrace \ge x) \ge \mathbb{P}(\max \lbrace |X_1|,\dots,|X_n|\rbrace \ge x) .$$ Thank you.
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$\begingroup$ Have you tried Slepian's inequality? en.wikipedia.org/wiki/Slepian%27s_lemma $\endgroup$– Liviu NicolaescuCommented Nov 26, 2015 at 17:03
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$\begingroup$ Thank you for the answer, I have tried Slepian inequality but $\Gamma_Y-\Gamma_X$ does not imply that the correlations of $Y$ are uniformly greater than correlation of $X$, and it seems to be difficult to adapt this lemma. $\endgroup$– Patrick TardivelCommented Nov 28, 2015 at 19:06
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I think that what you are writing is precisely Anderson's correlation inequality, see "T. W. Anderson. The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc., 6(2):170–176, 1955."
