On Mardia's kurtosis test for multivariate normality

Let X$_1$,..., X$_n$ be independent identically distributed random(d-vectors with mean μ and nonsingular covariance matrix $\Sigma$ such that E[{(X$_1$ - $\mu$)'$\Sigma$$^{-1}$(X$_1$ - $\mu$)}$^4$] < $\infty$. We show that Mardia’s measure b$_{2,d}$ = n$^{-1}$ $\Sigma$$_{j=1}^n$ {(X$_j$ - X$_n$)'S$_n^{-1}$ (X$_j$ - X$_n$)}$^2$ of multivariate kurtosis satisfies $\sqrt{n}$(b$_{2,b}$ - E[{(X$_1$ - $\mu$)'$\Sigma$$^{-1}$ (X$_1$ - $\mu$)}$^2$]) $\rightarrow$ N(0, $\sigma^2$) with $\sigma^2$ depending on the distribution of X$_1$. As a consequence we obtain an approximation to the power function of a commonly proposed test for multivariate normality based on d$_{2,d}$ . Moreover, this test is consistent if, and only if E[{(X$_1$ - $\mu$)'$\Sigma$$^{-1}$(X$_1$ - $\mu$)}$^2$] $\neq$ d(d + 2). Examples include normal mixtures and elliptically symmetric distributions.