Accurate Small Tail Probabilities of Sums of iid Lattice-Valued Random Variables via FFT

Accurately computing very small tail probabilities of a sum of independent and identically distributed lattice-valued random variables is numerically challenging. The only general purpose algorithms that can guarantee the desired accuracy have a quadratic runtime complexity that is often too slow. While fast Fourier transform (FFT)-based convolutions have an essentially linear runtime complexity, they can introduce overwhelming roundoff errors. We present sisFFT (segmented iterated shifted FFT), which harnesses the speed of FFT while retaining control of the relative error of the computed tail probability. We rigorously prove the method’s accuracy and we empirically demonstrate its significant speed advantage over existing accurate methods. Finally, we show that sisFFT sacrifices very little, if any, speed when FFT-based convolution is sufficiently accurate to begin with. Supplementary material is available online.