Let $\mathbf{X}_p = (X_{p,1}, \ldots, X_{p,p})$ follow the $p$-variate zero-mean Gaussian distribution $N_p(\mathbf{0}_p, \boldsymbol{\Sigma}_p)$. Then, the expectation and variance of the squared $L_2$-norm of $\mathbf{X}_p$, $\| \mathbf{X}_p \|_2^2$, is quite easy to compute using the Karhunen-Loeve expansion. But, how to compute the expectation and variance of the $L_\infty$-norm of $\mathbf{X}_p$, i.e., $\| \mathbf{X}_p \|_\infty = \max\{ | X_{p,j} | : j = 1, \ldots, p \}$, and what would be their values in terms of $\boldsymbol{\Sigma}_p$ or its eigenvalues?
Any direction in this would be much appreciated.
Is it possible to provide useful lower and upper bounds on the expectation and variance of $\| \mathbf{X}_p \|_\infty$ in terms of $\boldsymbol{\Sigma}_p$ and $p$?