Suppose $X \sim N(0, \Sigma)$ is a $d$-dimensional Gaussian random vector. And we have $2n$ $i.i.d$ sample $X_1, \ldots, X_{n}, \ldots, X_{2n}$.

Let $\hat{\Sigma}_1 = \frac{1}{n}\sum_{i=1}^nX_i X_i^\top$ and $\hat{\Sigma}_2 = \frac{1}{n}\sum_{i=n+1}^{2n}X_i X_i^\top$ be the sample covariance matrix of the first half and second half data.

For the first one, I use it to solve the following optimization problem:

for $i = 1:d$, \begin{equation} \min_{w_i \in R^{d}} \quad||w_i||_2 \\ \text{subject to } \quad||w_i^\top \hat{\Sigma}_1-e_i^\top||_{\infty} \leq \mu, \end{equation} where $e_i$ is the $i$-th basis in $R^d$.

Let $W \in R^{d \times d} = [w_1, w_2, \ldots, w_d]^\top$. How could I get a high probability bound that \begin{equation} \|W\hat{\Sigma}_2 - I\|_{\max} \leq C\mu, \end{equation} where $C$ is a generic constant.