# Efficient Estimation for Heavy-tailed Random Sums of Light-Tailed Random Variables

Random sums of random variables arise in many contexts of applied probability such as the probability of buffer overflow in queues, or when considering the probability of ruin for risk models. When such quantities of interest cannot be evaluated precisely, there are many well established techniques for estimating them via Monte Carlo methods: for example, via importance sampling using exponential twisting in the case of light-tailed sums of light-tailed random variables, or via the Asmussen--Kroese estimator in the case of light- or heavy-tailed sums of sub-exponential random variables. However, the estimation techniques for light- and heavy-tailed random sums are of quite different character.

This project focuses on the little-studied case of heavy-tailed sums of light-tailed random variables. Preliminary work suggests that existing techniques for estimating buffer overflow probabilities

do not perform well in this case. The aim of this project is to develop efficient techniques for estimating buffer overflow probabilities for heavy-tailed sums of light-tailed random variables, with a view toward

developing a class of estimation techniques for the buffer overflow problem that perform efficiently for both light- and heavy-tailed random sums.