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

Percentiles of sums of heavy-tailed random variables: Beyond the single-loss approximation