Let $X_1, X_2, \dots, X_n \sim X$ be i.i.d. random vectors in $\mathbb{R}^d$ with $\mathbb{E}[XX^{\top}] = \Lambda$. Let $\hat{\Lambda} = \sum_{j = 1}^{n} X_j X_j^{\top}$ be the empirical covariance matrix. There are several known results about the convergence of $\hat{\Lambda}$ to $n\Lambda$ in different norms as $n$ grows to infinity.
I was wondering if there any results that show the convergence of $(\hat{\Lambda} + I_d)^{-1}$ to $(n{\Lambda} + I_d)^{-1}$ or if one can show this convergence using the convergence of $\hat{\Lambda}$ to $n\Lambda$. Here $I_d$ refers to the $d$ dimensional identity matrix.
I am eventually interested in the case when the random vectors are infinite dimensional. However, the results for the finite dimensional case should be a good start for me. Any leads, hints or references will be appreciated. Thanks!