  We consider two nonparametric estimators for the risk measure of the sum of
$n$ i.i.d. individual insurance risks where the number of historical single
claims that are used for the statistical estimation is of order $n$. This
framework matches the situation that nonlife insurance companies are faced with
within in the scope of premium calculation. Indeed, the risk measure of the
aggregate risk divided by $n$ can be seen as a suitable premium for each of the
individual risks. For both estimators divided by $n$ we derive a sort of
Marcinkiewicz--Zygmund strong law as well as a weak limit theorem. The behavior
of the estimators for small to moderate $n$ is studied by means of Monte-Carlo
simulations.
