Non-Gaussian fluctuations for traces of squared sample correlation matrices in high dimensions

2026/08/06 by Johannes Heiny, Xuechun Hu, Felix Seo
Mathematics · #math.ST #math.PR #stat.TH #msc:60B20 #msc:60F05 #msc:60F10 #msc:60G55 #msc:60G70

paper · pdf

50 pages

arxiv created 2026/08/06 · arxiv updated 2026/08/07

Abstract

We provide limit theory for the trace of the squared sample correlation matrix \mathbf R, constructed from n observations of a p-dimensional random vector with iid components. If the entries have finite fourth moment and p and n grow proportionally, it is known that tr(\mathbf R²) satisfies a central limit theorem (CLT) and the centering and scaling sequences are universal in the sense that they do not depend on the entry distribution. Under symmetry and regular variation assumption with index α and any growth rate of the dimension, we prove that the universal CLT remains valid for α>3. For α<3, we identify a critical dimension growth at which the fluctuations of tr(\mathbf R²) become non-Gaussian. Moreover, if the dimension p grows faster and α≤ 3 we establish a non-universal CLT with norming sequences depending on the value of α. Our findings are illustrated in a simulation study.

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