Let $X_1, \dots, X_n$ be random real-valued symmetric rank-one matrices,
$$
X_i = x_i \otimes x_i,
$$
where $x_i$ are such that the standard Euclidean norm satisfies $\|x_i\|^2 \leq a$ almost surely.
Assume that $x_i$ are independently and identically distributed and let
$M = \mathbb{E}[X_i] = \mathbb{E}[x_i \otimes x_i]$, denote their common mean.
Define their average $\overline{X}_{n} = n^{-1} S_n$ where $S_n = \sum_{i=1}^n X_i$.
Let $f(T) := \mathrm{trace}((T + I)^{-1})$.
Question: What is the smallest constant $C = C_d(a, n) \geq 1$ such that we have $$ \mathbb{E}[f(\overline{X}_n)] \leq C~f(M)? $$
It should be emphasized that the constant $C$ is universal: it is valid for any law of $x_i$, supported on the Euclidean ball of (squared) radius $a$. It should be dependent only on $a, n$ and the dimension $d$.
Comments:
Necessarily $C \geq 1$. Note that by Jensen's inequality, we have the following inequality, $ \mathbb{E}[f(\overline{X}_n)] \geq f(M), $ since $f$ is a convex function on the symmetric positive definite matrices.
In the case $d = 1$, I was able to solve this problem and calculate the extremal distribution. See this post.