Let $x_1,\dots,x_n$ be i.i.d. $N(0,I_{p\times p})$, with $n>p$. Let $\hat S=\frac1n\sum_{i=1}^n x_i x_i^T$ be the sample covariance.

Assume the asymptotic setting where $\frac pn\to \alpha<1$.

Is there a result about the concentration of $\mathbb{E}\left[\operatorname{tr}\left(\hat{S}^{-1}\right)\right]$ in the asymptotic setting? By using Lemma 3.2 in this paper coupled with the Sherman-Morrison formula I believe that $\mathbb{E}\left[\operatorname{tr}\left(\hat{S}^{-1}\right)\right]=\frac{1}{1-\alpha}$, but I haven't been able to make it rigorous.

Are there any existing results about this?