Asymptotic theory for the test for multivariate normality by Cox and Small

We derive the limit distribution of the statistic of Cox and Small (1978) for testing multivariate normality when the underlying distribution is elliptically-symmetric. Moreover, we consider fixed and contiguous alternatives to normality. Empirical critical values as well as a Monte Carlo simulation for comparison to classical procedures are provided. We further show how some results can also be used for asymptotic results of the test for normality of Malkovich and Afifi.