Johannes Heiny: Recent advances in large sample correlation matrices and their applications

First, we consider a p-dimensional population with iid coordinates in the domain of attraction of a stable distribution with index α ∈ (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→ γ. The moments of the limiting distributions H_α,γ 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_α,γ} has continuous extensions at the boundaries α = 2 and α = 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.

In the second part of this talk, we assume that the coordinates of the p-dimensional population are dependent and p/n ≤ 1. Under a finite fourth moment condition on the entries we find that the log determinant of the sample correlation matrix R satisfies a central limit theorem. In the iid case, it turns out the central limit theorem holds as long as the coordinates are in the domain of attraction of a stable distribution with index α > 3, from which we conjecture a promising and robust test statistic for heavy-tailed high-dimensional data. The findings are applied to independence testing and to the volume of random simplices.