Many fields of modern sciences are faced with high-dimensional data sets. In this talk, we investigate the spectral properties of large sample correlation matrices.
First, 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.
In the second part of this talk, we assume that the coordinates of the $p$-dimensional population are dependent and $p/n \le 1$. 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 $\alpha>3$, 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.
Underlying papers:
J. Heiny, S. Johnston, J. Prochno: Thin-shell theory for rotationally invariant random simplices and
Personal Website of Johannes Heiny
The talk also can be joined online via our ZOOM MEETING
Meeting room opens at: March 14, 2022 4.30 pm Vienna
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