Limit laws for the norms of extremal samples

Let denote $S_n(p) = k_n^{-1} \sum_{i=1}^{k_n} \left( \log (X_{n+1-i,n} / X_{n-k_n, n}) \right)^p$, where $p > 0$, $k_n \leq n$ is a sequence of integers such that $k_n \to \infty$ and $k_n / n \to 0$, and $X_{1,n} \leq \ldots \leq X_{n,n}$ is the order statistics of iid random variables with regularly varying upper tail. The estimator $\widehat \gamma(n) = (S_n(p)/\Gamma(p+1))^{1/p}$ is an extension of the Hill estimator. We investigate the asymptotic properties of $S_n(p)$ and $\widehat \gamma(n)$ both for fixed $p > 0$ and for $p = p_n \to \infty$. We prove strong consistency and asymptotic normality under appropriate assumptions. Applied to real data we find that for larger $p$ the estimator is less sensitive to the change in $k_n$ than the Hill estimator.