Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations

Consider a $p$-dimensional population $\mathbf{x}\in {\mathbb{R}}^p$ with iid coordinates in the domain of attraction of a stable distribution with index $\alpha \in (0,2)$. Since the variance of $\mathbf{x}$ is infinite, the sample covariance matrix ${\mathbf S}n=n^{-1}\sum{i=1}^n {{\mathbf x}_i}{\mathbf x}'_i$basedonasample${\mathbf x}_1,\ldots,{\mathbf x}_n$ from the population is not well behaved and it is of interest to use instead the sample correlation matrix ${\mathbf R}_n= {\operatorname{diag}({\mathbf S}_n)}^{-1/2}, {\mathbf S}_n {\operatorname{diag}({\mathbf S}_n)}^{-1/2}$. This paper finds the limiting distributions of the eigenvalues of ${\mathbf R}_n$whenboththedimension$p$andthesamplesize$n$ grow to infinity such that $p\mathrm{/}n\to \gamma \in (0,\mathrm{\infty })$. The family of limiting distributions $\{H_{\alpha ,\gamma }\}$ is new and depends on the two parameters $\alpha$ and $\gamma$. The moments of $H_{\alpha ,\gamma }$ are fully identified as sum of two contributions: the first from the classical Mar\v{c}enko-Pastur law and a second due to heavy tails. Moreover, the family $\{H_{\alpha ,\gamma }\}$ has continuous extensions at the boundaries $\alpha =2$ and $\alpha =0$ leading to the Mar\v{c}enko-Pastur law and a modified Poisson distribution, respectively. Our proofs use the method of moments, the path-shortening algorithm developed in [18] and some novel graph counting combinatorics. As a consequence, the moments of $H_{\alpha ,\gamma }$ are expressed in terms of combinatorial objects such as Stirling numbers of the second kind. A simulation study on these limiting distributions $H_{\alpha ,\gamma }$ is also provided for comparison with the Mar\v{c}enko-Pastur law.