I am working on the limiting behavior for the eigenvalue and the corresponding eigenvectors, especially the minimum eigenvalues. For instance, let $S_X=\frac{1} {T} \sum_t X_t X_t ^\prime$ be a $p \times p$ sample covariance matrix, and $p$ is finite. Let $\lambda_i$ be the eigenvalues of $S_X$ such that $\lambda_1 \ge \lambda_2 \ge ...\ge \lambda_p$, and let $e_i$ be the corresponding eigenvector of $\lambda_i$ for $i=1,...,p$. The limiting behavior of both $\lambda_p$ and $e_p$ are well established if $\lambda_i$ is distinct for large $T$. However, my problem is whether the limiting behavior will change if the eigenvalues are not distinct. For instance, if $\lambda_p =\lambda_{p-1}$, then what's the limiting behavior of $e_p$ and $e_{p-1}$ as $T \to \infty $. Any related reference for this particular issue since my estimator depends on the eigenvectors corresponding the minimum eigenvalue? Thanks!


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