# Rate of convergence of the eigenvalues of the samples covariance matrix

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 $$\Theta_p$$?