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

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Many statistics are based on functions of sample moments. Important examples are the sample variance sₙ₋₁², 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 (∑ᵢ₌₁ⁿXᵢ,∑ᵢ₌₁ⁿXᵢ²) plays an important role. In studying the asymptotic behaviour of this vector we start by formulating best possible conditions under which the vector (X,X²) 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 t²(n). For simplicity, in restrict ourselves to positive random variables X.