Limit distributions of norms of vectors of positive i.i.d. random variables

This paper aims to combine the central limit theorem with the limit theorems in extreme value theory through a parametrized class of limit theorems where the former ones appear as special cases. To this end the limit distributions of suitably centered and normalized l(cp(n))-norms of n-vectors of positive i.i.d. random variables are investigated. Here, c is a positive constant and p(n) is a sequence of positive numbers that is given intrinsically by the form of the upper tail behavior of the random variables. A family of limit distributions is obtained if c runs over the positive real axis. The normal distribution and the extreme value distributions appear as the endpoints of these families, namely, for c = 0+ and c = infinity, respectively.