Limit laws for multivariate skewness in the sense of Mori, Rohatgi and Szekely

Let X be a d-dimensional random vector having zero expectation and unit covariance matrix. Mrri et al. (1993) proposed and studied $\beta_{d,1}$ = IE(IXI$^2$X)I$^2$ as a population measure of multivariate skewness. We derive the limit distribution of an affine invariant sample counterpart $b_{1,d}$, $\beta_{d,1}$. If the distribution of X is spherically symmetric, this limit law is ${\lambda_{Xd^2}}$, where $\lambda$ depends on EIXI$^4$ and EIXI$^6$. In case of spherical (elliptical) symmetry, we also obtain the asymptotic correlation between $b_{1,d}$ and Mardia's time-honoured measure of multivariate skewness. If $\beta_{1,d}$ > 0, the limit distribution of n$^{12}$(b$_{1,d}$ - $\beta_{d,1}$) is normal. Our results reveal the deficiencies of a test for multivariate normality based on b$_{1,d} \cdot$.