  The following class of sum-product statistics
  T_n(p)=\frac{1}{k}\sum_{h=1}^p \sum_{(s_1...s_h)\in P(p,h)}
\sum_{i_1=l+1}^{i_0} ... \sum_{i_h=l+1}^{i_{h-1}} i_h \prod_{i=i_1}^{i_h}
\frac{(Y_{n-i+1,n}-Y_{n-i,n})^{s_i}}{s_i!}
  (where $l,$ $k=i_{0}$ and n are positive integers, $0<l<k<n,$ $P(p,h)$ is the
set of all ordered parititions of $\ p>0$ into $\ h$ positive integers and
$Y_{1,n}\leq ...\leq Y_{n,n}$ are the order statistics based on a sequence of
independent random variables $Y_{1},$ $Y_{2},...$with underlying distribution
$\mathbb{P}(Y\leq y)=G(Y)=F(e^{y})$), is introduced. For each p,
$T_{n}(p)^{-1/p}$ is an estimator of the index of a distribution whose upper
tail varies regularly at infinity. \ This family generalizes the so called Hill
statistic and the Dekkers-Einmahl-De Haan one. We study the limiting laws of
the process ${T_{n}(p),1\leq p<\infty}$ and completely describe the covariance
function of the Gaussian limiting process with the help of combinatorial
techniques. Many results available for Hill's statistic regarding asymptotic
normality and laws of the iterated logarithm are extended to each margin
$T_{n}(p,k)$, for $p$ fixed, and for any distribution function lying in the
extremal domain. In the process, we obtain special classes of numbers related
to those of paths joining the opposite coins within a parallelogram.
