Nonsymmetric examples for Gaussian correlation inequalities

TL;DR

This paper investigates the variances of maxima of Gaussian variables with different covariance structures, proving a specific correlation inequality and showing that dependence increases variance compared to independence.

Contribution

It introduces a new covariance inequality for Gaussian maxima and demonstrates that dependence among variables increases the variance of their maximum.

Findings

Variance of maxima with correlated Gaussians exceeds that of independent ones.

Variance of maxima decreases as the number of i.i.d. Gaussian variables increases.

Established a specific convex/log-concave correlation inequality for Gaussian distributions.

Abstract

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 $N$.