  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. L\^{o}
(\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 :
  $$ k^{-1}\sum_{j=\ell +1}^{j=k}\ \sum_{i=j}^{i=k}i(1-\delta_{ij}/2)(\log
X_{n-i+1,n}-\log X_{n-i,n}) $$
  $$ \times (\log X_{n-j+1,n}-\log X_{n-j,n}t), $$
  \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
$\delta_{ij\text{}}$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)
