2021/12/31 by Johannes Heiny, Heiny, Johannes, Nestor Parolya +1 · 1 citation
Chemistry · Mathematics · Physics and Astronomy · #60B20 #60F05 #60G10 #60G57 #60G70 #Complex Network Analysis Techniques #FOS: Mathematics #Molecular spectroscopy and chirality #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2112.15388
openalex publication_date 2021/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we show the central limit theorem for the logarithmic determinant of the sample correlation matrix R constructed from the (p× n)-dimensional data matrix X containing independent and identically distributed random entries with mean zero, variance one and infinite fourth moments. Precisely, we show that for p/n→ γ∈ (0,1) as n,p→ ∞ the logarithmic law (log det R -(p-n+(1)/(2))log(1-p/n)+p-p/n)/(√(-2log(1-p/n)- 2 p/n)) \oversetd→ N(0,1) is still valid if the entries of the data matrix X follow a symmetric distribution with a regularly varying tail of index α∈ (3,4). The latter assumptions seem to be crucial, which is justified by the simulations: if the entries of X have the infinite absolute third moment and/or their distribution is not symmetric, the logarithmic law is not valid anymore. The derived results highlight that the logarithmic determinant of the sample correlation matrix is a very stable and flexible statistic for heavy-tailed big data and open a novel way of analysis of high-dimensional random matrices with self-normalized entries.