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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 $n<p$?

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For $n<p$ you have the socalled singular Wishart distribution. The matrix $M_s$ has $p-n$ eigenvalues identically equal to zero. See for example Chapter 4 of A Derivation of the Wishart and Singular Wishart Distributions.

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  • $\begingroup$ Thank you very much for your answer, is there any result about the local distribution of eigenvalues in this case (mean, variance, covariance)? $\endgroup$
    – SC_thesard
    Commented Jun 5, 2018 at 11:16
  • $\begingroup$ Just to make my question clear, there is an asympotic of the largest eigenvalue of Wishart matrix under the assumption $n>p$, is there any result similar when $n<p$? $\endgroup$
    – SC_thesard
    Commented Jun 5, 2018 at 11:21
  • $\begingroup$ I found a reference for this distribution in the complex case: arxiv.org/abs/0803.4155 --- and also in the real case: arxiv.org/abs/1203.0839 $\endgroup$
    – Carlo Beenakker
    Commented Jun 5, 2018 at 11:23

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