Domains of attraction of the real random vector $(X,X^2)$ and applications

Edward,Omey, Stefan,Van Gulck

Many statistics are based on functions of sample moments.Important examples are the sample variance $s^2(n)$, the sample coefficient of variation $SV(n)$,the sample dispersion $SD(n)$ and the non-central $t$-statistic $t(n)$.The definition of these quantities makes clear that the vector defined by$\big(\sum_{i=1}^n\!X_i^{},\,\sum_{i=1}^n\!X_i^2\big)$plays an important role.In the paper we obtain conditions under which the vector $(X,X^2)$belongs to a bivariate domain of attraction of a stable law.Applying simple transformations then leads to a full discussionof the asymptotic behaviour of $SV(n)$ and $t(n)$.