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

Abstract

Consider a p-dimensional population x∈Rp with i.i.d. coordinates that are regularly varying with index α∈(0,2). Since the variance of x is infinite, the diagonal elements of the sample covariance matrix Sn=n−1 ∑ i=1nxixi′ based on a sample x1,…,xn from the population tend to infinity as n increases and it is of interest to use instead the sample correlation matrix Rn={diag(Sn)}−1/2Sn{diag(Sn)}−1/2. This paper finds the limiting distributions of the eigenvalues of Rn when both the dimension p and the sample size n grow to infinity such that p/n→γ∈(0,∞). The family of limiting distributions {Hα,γ} is new and depends on the two parameters α and γ. The moments of Hα,γ 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 {Hα,γ} has continuous extensions at the boundaries α=2 and α=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 Hα,γ are expressed in terms of combinatorial objects such as Stirling numbers of the second kind. A simulation study on these limiting distributions Hα,γ is also provided for comparison with the Marčenko–Pastur law.