  Let $X$ be an $M\times N$ random matrix consisting of independent $M$-variate
elliptically distributed column vectors $\mathbf{x}_{1},\dots,\mathbf{x}_{N}$
with general population covariance matrix $\Sigma$. In the literature, the
quantity $XX^{*}$ is referred to as the sample covariance matrix after scaling,
where $X^{*}$ is the transpose of $X$. In this article, we prove that the
limiting behavior of the scaled largest eigenvalue of $XX^{*}$ is universal for
a wide class of elliptical distributions, namely, the scaled largest eigenvalue
converges weakly to the same limit regardless of the distributions that
$\mathbf{x}_{1},\dots,\mathbf{x}_{N}$ follow as $M,N\to\infty$ with
$M/N\to\phi_0>0$ if the weak fourth moment of the radius of $\mathbf{x}_{1}$
exists . In particular, via comparing the Green function with that of the
sample covariance matrix of multivariate normally distributed data, we conclude
that the limiting distribution of the scaled largest eigenvalue is the
celebrated Tracy-Widom law.
