The kurtosis of normal variance-mean mixtures

2026-06-22

AI Summary

This study investigates the nature of kurtosis in multivariate normal variance-mean mixture distributions, revealing that it arises from the interplay among mean variation, covariance structure, and the random mixing variable. By leveraging fourth-order cumulants, the paper presents the first decomposition of multivariate kurtosis into three interpretable components: directional, direction–covariance interaction, and covariance-pair terms, and establishes theoretical connections to Mardia’s kurtosis measure and projection pursuit. The proposed framework demonstrates that kurtosis is not merely a manifestation of tail behavior in specific directions but rather an aggregate signature of multiple sources of non-Gaussianity. Building on this insight, the authors develop novel diagnostic tools for assessing non-Gaussianity, identifying dominant tail directions, and detecting influential tail events, with empirical validation provided through simulations and daily stock return data.

Abstract

This paper studies kurtosis in multivariate normal variance-mean mixtures through its fourth-cumulant representation. We obtain an explicit expression for the fourth cumulant whose structure separates naturally into a rank-one directional component, a mixed direction--covariance component, and a covariance-pairing component induced by the mixing variable. This formulation shows that kurtosis in this class is not merely a directional tail phenomenon, but also reflects the interaction between mean variation, covariance structure, and stochastic mixing. We further derive the standardized fourth cumulant, relate it to Mardia's multivariate excess kurtosis, and study directional excess kurtosis through projection pursuit. Statistical applications are developed for cumulant-based diagnostics of multivariate non-Gaussianity, dominant-tail-direction analysis, and influential-tail-event detection. The practical relevance of the theoretical results is illustrated with simulated data and daily stock returns.

Problem

• kurtosis
• normal variance-mean mixtures
• fourth cumulant
• multivariate non-Gaussianity
• directional tail

Innovation

• kurtosis
• normal variance-mean mixtures
• fourth cumulant
• multivariate non-Gaussianity
• projection pursuit