Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations
share
Consider a p-dimensional population x∈R^p with iid coordinates in the domain of attraction of a stable distribution with index α∈ (0,2). Since the variance of x is infinite, the sample covariance matrix S_n=n^-1∑_i=1^n x_ix'_i based on a sample x_1,...,x_n from the population is not well behaved and it is of interest to use instead the sample correlation matrix R_n= {diag(S_n)}^-1/2 S_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→γ∈ (0,∞). The family of limiting distributions {H_α,γ} is new and depends on the two parameters α and γ. The moments of H_α,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 {H_α,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] and some novel graph counting combinatorics. As a consequence, the moments of H_α,γ are expressed in terms of combinatorial objects such as Stirling numbers of the second kind. A simulation study on these limiting distributions H_α,γ is also provided for comparison with the Marčenko-Pastur law.
READ FULL TEXT