Asymptotic analysis of a measure of variation. (English) Zbl 1150.62363

Albrecher, H.; Teugels, J. L.

Teor. Jmovirn. Mat. Stat. 74, 1-9 (2006).

This paper deals with the asymptotic behaviour of \(E[T^{k}_{n}]\), as \(n\to\infty\), where \[ T_{n}:=(X_1^2+X_2^2+\dots+X_{n}^2)(X_1+X_2+\dots+X_{n})^{-2}, \] and \(X_{i}\), \(i=1,\dots,n\), is a sequence of positive i.i.d. random variables with distribution function \(F\). The authors use an integral representation of \(E[T^{k}_{n}]\) in terms of the Laplace transform of \(X_{1}\). Most of the presented results are derived under the condition that \(X_1\) satisfies \(1-F(x)\sim x^{-\alpha}l(x)\), \(x\uparrow\infty\), where \(\alpha>0\) and \(l(x)\) is slowly varying function. For different values of \(\alpha\) the asymptotic behaviour of \(E[T^{k}_{n}]\) is studied. One of the presented results is the following:
If \(X_1\) belongs to the domain of attraction of a stable law with index \(0<\alpha<1\), then \[ \lim_{n\to\infty}E[T^{k}_{n}]=(k!/\Gamma(2k))\sum_{r=1}^{k}(\alpha^{r-1}/(r\Gamma(1-\alpha)^{r})G(r,k) \quad\text{for all}\;k\geq1, \] where \(G(r,k)\) is the coefficient of \(x^{k}\) in the polynomial \[ (\sum_{j=1}^{k-r+1}(\Gamma(2j-\alpha)/j!)x^{j})^{r}. \] A new method for estimating the extreme value index of Pareto-type distributions from a data set of observations is suggested.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

62G20	Asymptotic properties of nonparametric inference
62G32	Statistics of extreme values; tail inference

Keywords:

functions of regular variation; domain of attraction of stable laws; integral representation