Log determinant of large correlation matrices under infinite fourth moment

Abstract

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
_{→d N(0, 1)}
^√−2 log(1 − p/n) − 2p/n 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.