Nonsymmetric examples for Gaussian correlation inequalities

In this paper, we compare two variances of maxima of $N$ standard Gaussian random variables. One is a sequence of $N$ i.i.d. standard Gaussians, and the other one is $N$ standard Gaussians with covariances $\sigma_{1,2}=\rho \in(0,1)$ and $\sigma_{i,j}=0$, for other $i\neq j$. It turns out that we need to discuss the covariance of two functions with respect to multivariate Gaussian distributions. Gaussian correlation inequalities hold for many symmetric (with respect to the origin) cases. However, in our case, the max function and its derivatives are not symmetric about the origin. We have two main results in this paper. First, we prove a specific case for a convex/log-concave correlation inequality for the standard multivariate Gaussian distribution. The other result is that the variance of maxima of standard Gaussians with $\sigma_{1,2}=\rho \in(0,1)$, while $\sigma_{i,j}=0$, for other $i\neq j$, is larger than the variance of maxima of independent standard Gaussians. This implies that the variance of maxima of $N$ i.i.d. standard Gaussians is decreasing in $N$.