On the generalized Hill process for small parameters and applications

Let X1,X2,... be a sequence of independent copies (s.i.c) of a real random variable (r.v.) X> 1, with distribution function df F(x) = P(X 6 x) and let X1,n 6 X2,n 6 · · ·6 Xn,n be the order statistics based on the n> 1 first of these observations. The following continuous generalized Hill process Tn(τ) = k−τ j=k j=1 jτ logXn − j+1,n − logXn − j,n τ> 0, 1 6 k 6 n, has been introduced as a continuous family of estimators of the extreme value index, and largely studied for statistical purposes with asymptotic normality results restricted to τ> 1/2. We extend those results to 0 < τ 6 1/2 and show that asymptotic normality is still valid for τ = 1/2. For 0 < τ < 1/2, we get non Gaussian asymptotic laws which are closely related to the Riemann function ζ (s) = ∑∞n=1 n −s, s> 1