Assume $\{ x_i \}_{i=1}^n$ to be i.i.d. normally distributed with mean 0 and covariance matrix $\Sigma$.

What can we say about the convergence of the eigenvalues of the samples covariance matrix $\Sigma_n = \frac1n \sum_{i=1}^{n}x_i x_i^t$ to the eigenvalues of $\Sigma$?

I found this thread, but can we say something more when the data is normal? For example, to replace the $O_p$ notation to $\Theta_p$?



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