Efficient splitting-based rare event simulation algorithms for heavy-tailed sums

Rare events in heavy-tailed systems are challenging to analyze using splitting algorithms because large deviations occur suddenly. So, every path prior to the rare event is viable and there is no clear mechanism for rewarding and splitting paths that are moving towards the rare event of interest. We propose and analyze a splitting algorithm for the tail distribution of a heavy-tailed random walk. We prove that our estimator achieves the best possible performance in terms of the growth rate of the relative mean squared error, while controlling the population size of the particles.