Some inequalities between measures of multivariate kurtosis, with application to financial returns

The kurtosis of a random variable is often measured by its fourth standardized moment. Similarly, measures of multivariate kurtosis are often functions of a matrix containing all the fourth order moments which can be obtained from a standardized random vector. This paper examines some properties of the fourth moment matrix, and uses them to establish some inequalities between well-known scalar measures of multivariate kurtosis. Theoretical results are applied to multivariate financial returns.