Maximizing expectation of gaussian process over covariance matrix with fixed trace

Question

Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded.

For each $\Sigma \in \mathcal{A}$, let $X_{\Sigma} \sim N(0, \Sigma). $

Let $f_{1}(\Sigma):=\mathbb{E}[\max_{1 \leq i\leq n}(X_{\Sigma})_{i}]$, and let $f_{2}(\Sigma):=\mathbb{E}[\max_{1 \leq i\leq n}|(X_{\Sigma})_{i}|]$. Both $f_{1}$ and $f_{2}$ are convex functions on $\mathcal{A}$. By the extreme value theorem they attain a maximum and minium on $\mathcal{A}$. How can the extremes of $f_{1}$ and $f_{2}$ be characterized?

Answer 1

1. The maximizer over $\Sigma$'s with nonnegative entries is $\Sigma=I_n$.

This follows from Slepian's lemma and the formula $$EZ=\int_0^\infty dz\,P(Z>z)-\int_{-\infty}^0 dz\,P(Z<z), \tag{1}\label{1}$$ which holds if $Z$ is a random variable with $E|Z|<\infty$.

Indeed, by Slepian's lemma, if $(X_1,\dots,X_n)$ and $(Y_1,\dots,Y_n)$ are zero-mean Gaussian random vectors with $EX_i^2=EY_i^2$ for all $i$ and $EX_i X_j\le EY_i Y_j$ for all distinct $i,j$, then $P(\max_i X_i>z)\ge P(\max_i Y_i>z)$ for all real $z$ and, equivalently, $P(\max_i X_i<z)\le P(\max_i Y_i<z)$ for all real $z$. So, the claimed result follows immediately from \eqref{1}.

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2. If the $X_i$'s are exchangeable, then, similarly, the maximizer is the $\Sigma$ with all off-diagonal entries equal $-\frac1{n-1}$ (for $n\ge2$).

Indeed, in this case $\Sigma=\frac1{1-c}(I_n-c1_n1_n^\top)$ for some real $c\ne1$, where $1_n:=[1,\dots,1]^\top$, and the eigenvalues of $\Sigma$ are $\frac1{1-c}$ (of multiplicity $n-1$) and $\frac{1-cn}{1-c}$. So, $\Sigma$ is positive semidefinite iff $c\le\frac1n$ iff the off-diagonal entries $-\frac c{1-c}$ of $\Sigma$ are $\ge-\frac1{n-1}$. So, the claim for exchangeable $X_i$'s follows by \eqref{1} and Slepian's lemma.

Comment: It follows from the answer by Guillaume Aubrun that, remarkably, the exchangeability condition can be dropped -- in view of a recent proof of the longstanding simplex mean width conjecture (of which I had not known).

Answer 2

I assume all random variables are centered. It is clear that $f_1$ and $f_2$ are minimized when $\Sigma_{ij}=1$, i.e., the Gaussian vector is distributed as $(X,X,\dots,X)$ with $X \sim N(0,1)$.

The maximization problem is more interesting. The same question has been asked already some years ago: Maximum of the expectation of maximum of Gaussian variables

The fact that $f_2$ is maximized for $\Sigma = \mathrm{Id}$ follows from the Khatri-Sidak lemma. The fact that $f_1$ is maximized for $\Sigma_{ij} = -1/(n-1)$ is a long-standing conjecture known as the simplex mean width conjecture ; see www.math.ualberta.ca/~alexandr/OrganizedPapers/AL-simp-wv.pdf for many equivalent formulations.

A solution to this conjecture has been announced in https://arxiv.org/abs/2112.03393