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

Abstract

Consider a p-dimensional population $\mathbf{x}\in {\mathbb{R}}^{\mathit{p}}$ with i.i.d. coordinates that are regularly varying with index $\mathit{\alpha }\in (0,2)$. Since the variance of x is infinite, the diagonal elements of the sample covariance matrix ${\mathbf{S}}_{\mathit{n}}={\mathit{n}}^{-1}{\sum }_{\mathit{i}=1}^{\mathit{n}}{\mathbf{x}}_{\mathit{i}}{\mathbf{x}}_{\mathit{i}}^{\prime }$ based on a sample ${\mathbf{x}}_1,\dots ,{\mathbf{x}}_{\mathit{n}}$ from the population tend to infinity as n increases and it is of interest to use instead the sample correlation matrix ${\mathbf{R}}_{\mathit{n}}={\{diag({\mathbf{S}}_{\mathit{n}})\}}^{-1/2}{\mathbf{S}}_{\mathit{n}}{\{diag({\mathbf{S}}_{\mathit{n}})\}}^{-1/2}$. This paper finds the limiting distributions of the eigenvalues of ${\mathbf{R}}_{\mathit{n}}$ when both the dimension p and the sample size n grow to infinity such that $\mathit{p}/\mathit{n}\to \mathit{\gamma }\in (0,\infty )$. The family of limiting distributions $\{{\mathit{H}}_{\mathit{\alpha },\mathit{\gamma }}\}$ is new and depends on the two parameters α and γ. The moments of ${\mathit{H}}_{\mathit{\alpha },\mathit{\gamma }}$ are fully identified as sum of two contributions: the first from the classical Marčenko–Pastur law and a second due to heavy tails. Moreover, the family $\{{\mathit{H}}_{\mathit{\alpha },\mathit{\gamma }}\}$ has continuous extensions at the boundaries $\mathit{\alpha }=2$ and $\mathit{\alpha }=0$ leading to the Marčenko–Pastur law and a modified Poisson distribution, respectively.

Our proofs use the method of moments, the path-shortening algorithm developed in [18] (Stochastic Process. Appl. 128 (2018) 2779–2815) and some novel graph counting combinatorics. As a consequence, the moments of ${\mathit{H}}_{\mathit{\alpha },\mathit{\gamma }}$ are expressed in terms of combinatorial objects such as Stirling numbers of the second kind. A simulation study on these limiting distributions ${\mathit{H}}_{\mathit{\alpha },\mathit{\gamma }}$ is also provided for comparison with the Marčenko–Pastur law.