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Convergence of Eigenvalues

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

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Do you mean $A_n = \frac{1}{n} \sum_{i=1}^n X_i X_i^T$? Also, presumably the $X_i$ are independent. – Robert Israel Dec 14 2011 at 18:51
There's a lot of literature on this problem. I suggest starting by taking a look at this paper: www-personal.umich.edu/~romanv/papers/… and other recent papers by Roman Vershynin. – Mark Meckes Dec 14 2011 at 19:28
@ Robert : Yes I do mean $A_n = \frac{1}{n}\sum_{i = 1}^{n}X_i X_i^T$. Sorry about the typo. – Ashin Dec 14 2011 at 19:59
@Ashin: then perhaps you should edit the question accordingly. – Ori Gurel-Gurevich Dec 15 2011 at 0:45