Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, and define the $d \times d$ psd matrix $S$ by
$$ S = \frac{1}{n}\sum_{i \in [n],\, s_i \ge q_b} x_ix_i^\top, $$ where $q_b=s_{(nb)}$ is the value of the $nb$th largest value of $s_1,\ldots,s_n$, where $s_i := |x_i^\top v|$.
Question. As a function of $a$ and $b$, what is the the value of $\ell := \lim_{n \to \infty} \mathbb E\, \operatorname{tr} S^{-1}$ ?
N.B. I'm fine with fixed-point equations for $\ell$.
A Trivial Example
Note that if $b=0$, then $q_b=q_0=\min_i s_i$, and so $S = (1/n)\sum_{i=1}^n x_i x_i^\top$. We deduce that $$ \ell := \lim_{n \to \infty} \mathbb E\, \operatorname{tr} S^{-1} = \lim_{n \to \infty} \frac{\operatorname{tr}\mathrm{cov}(x_1)}{n-d-2} = \lim_{n \to \infty} \frac{d}{n-d-2} = \frac{a}{1-a}. $$
