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 $c = \infty$,
respectively.