Simple derivation of Muirhead kurtosis for a multivariate t distribution

Muirhead’s kurtosis is not well known because it is hard to calculate the first and the second derivatives of the characteristic functions of corresponding distributions. Besides, evaluation of the characteristic functions itself is not easy to do. Especially we devise two easy ways based on stochastic representation of elliptical distributions to calculate Muirhead’s kurtosis for elliptical distributions and applied to a multivariate t distribution which is a subclass of elliptical distributions.