Eigenvalues of a high dimensional sample covariance matrix

Question

I am not a math/stat major, but I encountered high dimensional matrix in my research. Particularly, consider a sample covariance matrix of the form $S_{X}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{X}_{t}\mathbf{X}_{t}^{T},$ where $\mathbf{X}_{t}=\left( x_{1t},\ldots ,x_{Nt}\right) ^T.$
Assuming both $N$ and $T$ are large, then $S_{X}$ is a $N\times N$ high dimensional sample covariance matrix. Since the elements in $\mathbf{X}_{t}$ could be correlated, $S_{X}$ may not be a diagonal matrix. My question is what's the eigenvalues of $S_{X}$? Are there any references to find the properties of the eigenvalues of $S_{X}?$ The results I need is that the maximum eigenvalues of $S_{X}$ is bounded for large $N$ and $T.$ Thanks!

Answer 1

With normal 0 mean assumptions, $TS_x$ is Wishart distributed and the joint distribution of eigenvalues is known in some case (see wiki). Some result about the dominant eigenvalue distribution was derived in

Khatri, C. G. "Non-central distributions of i th largest characteristic roots of three matrices concerning complex multivariate normal populations." Annals of the Institute of Statistical Mathematics 21.1 (1969): 23-32.

Also this

Chiani, Marco. "Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution." Journal of Multivariate Analysis 129 (2014): 69-81..

You might start here.