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

Abstract

Many statistics are based on functions of sample moments. Important examples are the sample variance , 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

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.

Keywords:

• domains of attraction
• regular variation
• sample dispersion
• sample coefficient of variation
• non-central t-statistic

AMS Subject Classification:

• Primary: 60E05
• 60F05
• Secondary: 62E20
• 91B70