A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random Variables

TL;DR

This paper introduces a deterministic numerical method using Newton-Cotes rules for efficiently and accurately computing rare event probabilities in the left tail of sums of non-negative independent random variables, outperforming FFT in error preservation.

Contribution

The paper develops a flexible, robust numerical approach based on iterative convolution and Newton-Cotes rules, with comprehensive error analysis and comparison to FFT, applicable to various non-negative continuous RVs.

Findings

Newton-Cotes rules preserve relative error better than FFT.

The method is efficient and flexible for different types of non-negative RVs.

Numerical experiments confirm the theoretical advantages and robustness of the approach.

Abstract

We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons-Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton-Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency-benefits to using FFT, Newton-Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method's performance and support our theoretical findings.