  A perturbative approach is used to derive approximations of arbitrary order
to estimate high percentiles of sums of positive independent random variables
that exhibit heavy tails. Closed-form expressions for the successive
approximations are obtained both when the number of terms in the sum is
deterministic and when it is random. The zeroth order approximation is the
percentile of the maximum term in the sum. Higher orders in the perturbative
series involve the right-truncated moments of the individual random variables
that appear in the sum. These censored moments are always finite. As a result,
and in contrast to previous approximations proposed in the literature, the
perturbative series has the same form regardless of whether these random
variables have a finite mean or not. The accuracy of the approximations is
illustrated for a variety of distributions and a wide range of parameters. The
quality of the estimate improves as more terms are included in the perturbative
series, specially for higher percentiles and heavier tails.
