Given a true covariance matrix $M$ of dimension $p \times p$, we generate $n$ gaussian random vectors $X_1,..X_n \sim N(0,M)$. We then get a sample covariance matrix $M_s$ based on these $n$ realisations. What is the local joint or marginal distribution of $M_s$'s eigenvalues (in order)? For example, the distribution of the largest and second largest eigenvalues with their covariance?

When the number of observations $n$ is larger than the dimension $p$, the marginal distribution of local eigenvalues follows the Wishart distribution. https://en.wikipedia.org/wiki/Wishart_distribution What happens if $n<p$?