  We suggest approximating the distribution of the sum of independent and
identically distributed random variables with a Pareto-like tail by combining
extreme value approximations for the largest summands with a normal
approximation for the sum of the smaller summands. If the tail is well
approximated by a Pareto density, then this new approximation has substantially
smaller error rates compared to the usual normal approximation for underlying
distributions with finite variance and less than three moments. It can also
provide an accurate approximation for some infinite variance distributions.
