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

Abstract

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. For high percentiles, and specially for heavier tails, the quality of the estimate improves as more terms are included in the series, up to a certain order. Beyond that order the convergence of the series deteriorates. Nevertheless, the approximations obtained by truncating the perturbative series at intermediate orders are remarkably accurate for a variety of distributions in a wide range of parameters.