TITLE: Inequality for the maximum of Gaussian variables
QUESTION [7 upvotes]: 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.

REPLY [6 votes]: 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."