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

Many statistics are based on functions of sample moments. Important examples are the sample variance $s_{n-1}^2$, 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 (\sum_{i=1}^nX_i,\sum_{i=1}^nX_i^2) plays an important role. In studying the asymptotic behaviour of this vector we start by formulating best possible conditions under which the vector $(X,X^2)$ belongs to a bivariate domain of attraction of a stable law. This approach is new, uniform and simple. Our main results include a full discussion of the asymptotic behaviour of SV(n), SD(n) and $t^2(n)$. For simplicity, in restrict ourselves to positive random variables $X$.