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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 theset 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 ofindependent 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 uppertail varies regularly at infinity. \ This family generalizes the so called Hillstatistic and the Dekkers-Einmahl-De Haan one. We study the limiting laws ofthe process ${T_{n}(p),1\leq p<\infty}$ and completely describe the covariancefunction of the Gaussian limiting process with the help of combinatorialtechniques. Many results available for Hill's statistic regarding asymptoticnormality 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 theextremal domain. In the process, we obtain special classes of numbers relatedto those of paths joining the opposite coins within a parallelogram.

On a discrete Hill's statistical process based on sum-product statistics and its finite-dimensional asymptotic theory