Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations
The Annals of Statistics · 2022 · Volume 50 · Issue 6
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.
Keywords
60B20, 60F05, 60G10, 60G57, 60G70, infinite variance, Limiting spectral distribution, Marčenko–Pastur law, method of moments, sample correlation matrix, stable distribution
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