Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations
Abstract
We consider a p-dimensional population with iid coordinates in the domain of attraction of a stable distribution with index $\alpha\in (0,2)$. Since
the variance is infinite, the sample covariance matrix based on a sample of size n from the population is not well behaved and it is of interest to
use instead the sample correlation matrix R. We find the limiting distributions of the eigenvalues of R when both the dimension p and the sample
size n grow to infinity such that $p/n\to \gamma$. The moments of the limiting distributions $H_{\alpha,\gamma}$ are fully identified as the sum
of two contributions: the first from the classical Marchenko-Pastur law and a second due to heavy tails. Moreover, the family
$\{H_{\alpha,\gamma}\}$ has continuous extensions at the boundaries $\alpha=2$ and $\alpha=0$ leading to the Marchenko-Pastur law and a
modified Poisson distribution, respectively. A simulation study on these limiting distributions is also provided for comparison with the
Marchenko-Pastur law. Our proofs use the method of moments, a path-shortening algorithm and some novel graph counting combinatorics.
The talk is based on joint work with Nina Dörnemann (Aarhus) and Jianfeng Yao (Shenzhen).
Reference: Heiny, J., and Yao, J. Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations. Annals of
Statistics 50 (2022), no. 6, 3249–3280.