Limit laws for the norms of extremal samples

Denote Sn(p)=kn−1∑i=1knlog(Xn+1−i,n∕Xn−kn,n)p, where p>0, kn≤n is a sequence of integers such that kn→∞ and kn∕n→0, X1,n≤⋯≤Xn,n are the order statistics iid random variables X1,…,Xn with regularly varying upper tail index 1∕γ. The estimator γ̂(n)=(Sn(p)∕Γ(p+1))1∕p an extension Hill estimator. We investigate asymptotic properties Sn(p) γ̂(n) both for fixed p>0 p=pn→∞. prove consistency limit theorem γ̂(n)−γ under appropriate assumptions. obtain Gaussian non-Gaussian (stable) depending on growth rate power pn. Applied to real data we find larger p less sensitive change in kn than