The kurtosis of normal variance-mean mixtures

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.