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

TL;DR

This paper investigates the asymptotic behavior of vectors formed by sample sums of positive random variables and their squares, establishing conditions for their convergence to stable laws and analyzing related statistical quantities.

Contribution

It introduces a new, uniform, and simple approach to determine when the vector (X, X^2) belongs to a bivariate domain of attraction of a stable law, with applications to common statistics.

Findings

Characterization of conditions for (X, X^2) to be in a stable law domain

Asymptotic analysis of sample variance, dispersion, and t-statistics

Simplified treatment for positive random variables

Abstract

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 $X$.