Let $X_{1},X_{2},...$ be a sequence of independent random variables ($rv$)with common distribution function ($df$) $F$ such that $F(1)=0$ and for each $n\geq 1,$ let $X_{1,n}\leq X_{2,n}\leq ...\leq X_{n,n}$ denote the order statistics based on the n first of these random variables. Lô (\cite{gslod}) introduced a class of statistics aimed at characterizing the asymptotic behavior of the univatiate extremes. This class this estimator of the square of the extremal index of a $df$ lying in the extremal domain of attraction :
$$ k^{-1}\sum_{j=\ell +1}^{j=k}\ \sum_{i=j}^{i=k}i(1-δ_{ij}/2)(\log X_{n-i+1,n}-\log X_{n-i,n}) $$
$$ \times (\log X_{n-j+1,n}-\log X_{n-j,n}t), $$
\noindent where $(k,\ell)$ is a couple of integers such that $k\rightarrow +\infty $, $k/n\rightarrow 0$, $\ell ^{2}/k\rightarrow 0$, as $n\rightarrow 0\rightarrow,$ $log$ stands for the natural logarithm and $δ_{ij\text{}}$is the Kronecker symbol.
\noindent In total $\mathbb{R}^{8}$-vectors are used in this paper and include the most popular statistics used in the literature. We consider here a multivariate approach and provide the asymptotic laws of such vectors. This allows quickly finding asymptotic laws of functional of new statistics and new estimators of the extremal index such as the Dekkers \textit{et al.} (1989) and Hasofer and Wang (1992) statistics for example as in Hah et al.(2012)
Centro de Matemática da Universidade do Porto（波尔图大学数学中心）；Faculdade de Economia da Universidade do Porto（波尔图大学经济学院）；Centro de Matemática & Faculdade de Ciências da Universidade do Porto（波尔图大学数学中心与理学院）；Mathematical Institute University of St Andrews（圣安德鲁斯大学数学研究所）
Local asymptotic normality in δ-neighborhoods of standard generalized Pareto processes
De Haan和Pereira（2006）提供了平稳性情况下的空间极值模型，该模型仅依赖于一个参数β > 0来衡量尾部依赖性，他们为该参数提出了不同的估计量。Falk（2011）通过建立高于高多元阈值的相应点过程的局部渐近正态性（LAN）对该框架进行了补充，特别是产生渐近有效估计。在这些论文中研究的估计量是基于点t1，.的有限集，td，在此进行观察。我们推广这种方法的背景下，功能极值理论（EVT）。这个更一般的框架允许在某个空间参数空间上进行估计，即，点t1，.的有限集在[a，b]中，td被t代替。特别地，我们基于C[0，1]中iid过程的样本中超过给定门限函数的过程，得到了β的有效估计.
