A large deviations approach to limit theory for heavy-tailed time series

• Published in: Probability theory and related fields Vol. 166; no. 1-2; pp. 233 - 269
• Main Authors: Mikosch, Thomas, Wintenberger, Olivier
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2016; Springer Nature B.V; Springer Verlag
• Subjects: Autoregressive models; Autoregressive processes; Central limit theorem; Deviation; Economics; Finance; Functionals; Insurance; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Multivariate analysis; Operations Research/Decision Theory; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Random walk; Statistics; Statistics for Business; Stochastic models; Stochasticity; Studies; Theoretical; Time series; Volatility; Large deviation principle; Central limit theorem; Ruin probabilities; 60G70; Secondary 60F05; GARCH; Regularly varying processes; Primary 60F10
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-015-0654-4

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.