Importance Sampling Strategy for Heavy-Tailed Systems with Catastrophe Principle

Large deviations theory has a long history of providing powerful machinery for designing efficient rare event simulation techniques. However, traditional large deviations theory fails to provide useful bounds in heavy-tailed contexts, and designing efficient rare-event simulation algorithms for heavy-tailed systems has been considered challenging. Recent developments in the theory of heavy-tailed large deviations enable designing a strongly efficient importance sampling scheme that is universally applicable to a wide range of rare events. This tutorial aims to provide an accessible overview of the recent developments in the large deviations theory for heavy-tailed stochastic processes, which is followed by a detailed account of the design principle behind the strongly efficient importance sampling scheme for such processes. The implementations of the general principle are demonstrated through a few specific heavy-tailed rare events that arise in stochastic approximation, finance, and queueing theory contexts.