Many rare events that arise in real-life applications exhibit heavy-tailed phenomena: for example, financial losses, delays in communication networks, and magnitudes of systemic events such as large-scale blackouts in power grids. While the theory of large deviations has been extremely successful in providing systematic tools for understanding rare events when the underlying uncertainties are light-tailed, the theory developed for the heavy-tailed counterparts has been mostly restricted to model-specific results or results pertaining to the events that are caused by a single big jump.
In this talk, I will discuss a new set of tools that goes far beyond such restrictions. Our new large deviations results can deal with a very general class of rare events associated with heavy-tailed random walks and Levy processes. In particular, we will fully characterize the "principle of (multiple) big jumps." Moreover, building on the sharp asymptotics provided by our new limit theorems, I will show how to construct simple, universal, and provably efficient rare-event simulation algorithms for heavy-tailed rare events, which has long been considered challenging. I will illustrate the application of the general theory with examples that arise in the context of mathematical finance, actuarial science, and queueing theory. Finally, I will also discuss how our new results provide an answer to an open question regarding the tail asymptotics of multiple server queues with heavy-tailed service times.