A large deviations approach to limit theory for heavy-tailed time series - Mikosch, Thomas; Wintenberger, Olivier (Probability theory and related fields, 2016)
In this paper we propagate a large deviations approach for proving limit theory for (generally) multivariate time series with heavy tails. We make this notion precise by introducing regularly varying time series. We provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin. We study a variety of such functionals, including large deviations of random walks, their suprema, the ruin functional, and further derive weak limit theory for maxima, point processes, cluster functionals and the tail empirical process. One of the main results of this paper concerns bounds for the ruin probability in various heavy-tailed models including GARCH, stochastic volatility models and solutions to stochastic recurrence equations.