Limit distributions for Mardia's measure of multivariate skewness

We study the asymptotic behavior of Mardia's measure of (sample) multivariate skewness. In the special case of an elliptically symmetric distribution, the limit law is a weighted sum of two independent χ$^2$-variates. A normal limit distribution arises if the population distribution has positive skewness. These results explain some curiosities in the power performance of a commonly proposed test for multivariate normality based on multivariate skewness.