Let $X \in \mathbb{R}^{n \times N}$ be a Gaussian random matrix with independent standard Normal entries; assume $N > n$. Fix a unit vector $u \in \mathbb{R}^{n}$. For a subset $S$ of the integers $[N] := \{1, \dots, N\}$ define $$ \Sigma_S = \frac{1}{|S|} \sum_{i \in S} X_i \otimes X_i \in \mathbb{R}^{n \times n} $$ where $\{X_i\}_{i=1}^N \subset \mathbb{R}^{n}$ denote the columns of $X$.
Define $$ F_{n,N}(u) = \mathbb{E} \bigg[\min_{\substack{S \subset [N] \\ |S|= n}} \langle \Sigma_S^{-1} u, u \rangle \bigg]. $$
I am wondering if we can establish the sharp behavior of this function.
Note that for any fixed $|S| = n$, we have $\mathbb{E} \langle \Sigma_S^{-1} u,u \rangle = \tfrac{1}{n}\mathbb{E}~ \mathrm{Tr}(\Sigma_S^{-1})$, which is infinite. I just note this to say that the minimum is essential. Of course, this also means our assumption $N > n$ is necessary to have $F_{n, N}(u) < \infty$; I am not sure if this is sufficient.
