Domains of attraction of the real random vector (X,X2) and applications

Abstract

Many statistics are based on functions of sample moments. Important examples are the sample variance s2(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 (?ni=1Xi, ?ni=1Xi2)plays an important role. In the paper we obtain conditions under which the vector (X,X2) belongs to a bivariate domain of attraction of a stable law. Applying simple transformations then leads to a full discussion of the asymptotic behaviour of SV(n) and t(n).