Suppose we have a matrix $A_n = \frac{1}{n}\sum_{i=1}^nX_i X_i^T$, where $X_i$ is a $p$-dimensional random-vector. We also know that $E(XX^T) = \Sigma_{p \times p}$. Let us denote the $j$-th largest eigenvalue of a symmetric positive-definite matrix $M$ by $\lambda_j(M)$. Then can we say anything about convergence of $\lambda_j(A_n) \rightarrow \lambda_j(\Sigma)$ as $n \rightarrow \infty$, that is, whether it converges in probability or in distribution and if so can we characterize the rate of convergence.

Thanks a lot, any help is much appreciated.

Best Ashin