LIMITING DISTRIBUTIONS FOR EIGENVALUES OF SAMPLE CORRELATION MATRICES FROM HEAVY-TAILED POPULATIONS

Consider a p-dimensional population x is an element of R-p with i.i.d. coordinates that are regularly varying with index alpha is an element of(0, 2). Since the variance of x is infinite, the diagonal elements of the sample covariance matrix S-n = n(-1) Sigma(n)(i=1) x(i)x(i)' based on a sample x(1), . . . , x(n) from the population tend to infinity as n increases and it is of interest to use instead the sample correlation matrix R-n = {diag(S-n)}S--1/2(n){diag(S-n)}(-1/2). This paper finds the limiting distributions of the eigenvalues of R-n when both the dimension p and the sample size n grow to infinity such that p/n -> gamma is an element of(0, infinity). The family of limiting distributions {H-alpha,H- gamma} is new and depends on the two parameters alpha and gamma. The moments of H-alpha,H- gamma are fully identified as sum of two contributions: the first from the classical Marcenko-Pastur law and a second due to heavy tails. Moreover, the family {H-alpha,H- gamma} has continuous extensions at the boundaries alpha = 2 and alpha = 0 leading to the Marcenko-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 H-alpha,H- 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,H- gamma is also provided for comparison with the Marcenko-Pastur law.