  The tail of the distribution of a sum of a random number of independent and
identically distributed nonnegative random variables depends on the tails of
the number of terms and of the terms themselves. This situation is of interest
in the collective risk model, where the total claim size in a portfolio is the
sum of a random number of claims. If the tail of the claim number is heavier
than the tail of the claim sizes, then under certain conditions the tail of the
total claim size does not change asymptotically if the individual claim sizes
are replaced by their expectations. The conditions allow the claim number
distribution to be of consistent variation or to be in the domain of attraction
of a Gumbel distribution with a mean excess function that grows to infinity
sufficiently fast. Moreover, the claim number is not necessarily required to be
independent of the claim sizes.
