# Limiting results for non-unique eigenvalues and eigenvectors for a sample covariance matrix

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!