THE NUMBER AND SUM OF NEAR m–EXTREMES

Abstract

Let $\{X_{n,}n\ge 1\}$ be a sequence of independent random variables with common continuous distribution function $F$. In this article we discuss distributional and asymptotical properties of the point process $N_{n,m}(·)=\sum _{i=1}^{n}1(X_{n-m+1:n}-X_i\in ·)$ driven by the $m$th upper order statistic $X_{n-m+1:n}$. Further we derive some limiting results for related sums, which are of some interest in insurance applications.