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Let $S_1$ and $S_2$ be the collection of i.i.d. copies of $X\sim\mathcal{N}(0,I_p)$, where $|S_1|=n_1,|S_2|=n_2$. Let $\hat{\Sigma}_1$ and $\hat{\Sigma}_2$ be the covariance matrix using samples in $S_1$ and $S_2$ respectively. Assume $p/n_1\rightarrow\gamma_1,p/n_2\rightarrow\gamma_2$. What I want to know is the limit spectral distribution of the random matrix $(\hat{\Sigma}_1+\hat{\Sigma}_2)^{-1}\hat{\Sigma}_1$ as $p,n_1,n_2$ tends to infinity. Or at least I hope to get the limit quantity $tr((\hat{\Sigma}_1+\hat{\Sigma}_2)^{-1}\hat{\Sigma}_1)$.

I am new to random matrix theory. As far as I know, suppose $S_1$ or $S_2$ is empty, then it is a standard result from this paper. However, if they are both non-empty, it seems nontrivial to obtain the limit spectral distribution.

I would be grateful if someone can give the result or provide me with some useful references, thank you!

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  • $\begingroup$ It seems like a usual quantity to asses equality of covariance matrices. The book "Sample covariance matrices and high-dimensional data analysis" by Yao, Zheng and Bai or the book "An introduction to multivariate statistics" by Anderson might contain such results. $\endgroup$
    – Gilles Mordant
    May 2 at 19:45

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