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

TL;DR

This paper introduces a perturbative method to accurately estimate high percentiles of sums of heavy-tailed random variables, improving upon previous approximations by incorporating higher-order terms and right-truncated moments.

Contribution

It develops a systematic perturbative series for percentile estimation that remains valid regardless of the finiteness of the mean, with closed-form expressions for deterministic and random sum sizes.

Findings

Higher-order approximations improve accuracy for high percentiles.

The method applies to distributions with infinite and finite means.

Approximation quality increases with more perturbative terms.

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. 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.