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In this paper, we show the central limit theorem for the logarithmicdeterminant of the sample correlation matrix $\mathbf{R}$ constructed from the$(p\times n)$-dimensional data matrix $\mathbf{X}$ containing independent andidentically distributed random entries with mean zero, variance one andinfinite fourth moments. Precisely, we show that for $p/n\to \gamma\in (0,1)$as $n,p\to \infty$ the logarithmic law \begin{equation*} \frac{\log \det\mathbf{R} -(p-n+\frac{1}{2})\log(1-p/n)+p-p/n}{\sqrt{-2\log(1-p/n)- 2 p/n}}\overset{d}{\rightarrow} N(0,1)\, \end{equation*} is still valid if the entriesof the data matrix $\mathbf{X}$ follow a symmetric distribution with aregularly varying tail of index $\alpha\in (3,4)$. The latter assumptions seemto be crucial, which is justified by the simulations: if the entries of$\mathbf{X}$ have the infinite absolute third moment and/or their distributionis not symmetric, the logarithmic law is not valid anymore. The derived resultshighlight that the logarithmic determinant of the sample correlation matrix isa very stable and flexible statistic for heavy-tailed big data and open a novelway of analysis of high-dimensional random matrices with self-normalizedentries.

Log determinant of large correlation matrices under infinite fourth moment