Limit Theorems on Order Statistics

Abstract

Let $F$ belong to the domain of attraction of a stable law with parameters $\alpha$ and $p$. Let $X_1, X_2, \cdots$ be a sample from $F$. Put $|\tilde X_1| \leq |\tilde X_2| \leq \cdots \leq |\tilde X_n|$. We consider the asymptotic properties as $n \rightarrow \infty$ (and $k \rightarrow \infty$) of the ratio of order statistics $(\tilde X_1 + \cdots + \tilde X_{n - k})/|\tilde X_{n - k + 1}|$.