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

Abstract

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. 1. Preliminaries and basic motivation In the last decades, a lot of efforts has been put into the understanding of limit theory for dependent sequences, including Markov chains (Meyn and Tweedie [42]), weakly dependent sequences (Dedecker et al. [21]), long-range dependent sequences (Doukhan et al. [23], Samorodnitsky [54]), empirical processes (Dehling et al. [22]) and more general structures (Eberlein and Taqqu [25]), to name a few references. A smaller part of the theory was devoted to limit theory under extremal dependence for point processes, maxima, partial sums, tail empirical processes. Resnick [49, 50] started a systematic study of the relations between the convergence of point processes, sums and maxima, see also Resnick [51] for a recent account. He advocated the use of multivariate regular variation as a flexible tool to describe heavy-tail phenomena combined with advanced continuous mapping techniques. For example, maxima and sums are understood as functionals acting on an underlying point process, if the point process converges these functionals converge as well and their limits are described in terms of the points of the limiting point process. Davis and Hsing [13] recognized the power of this approach for limit theory of point processes, maxima, sums, and large deviations for dependent regularly varying processes, i.e., stationary sequences whose finite-dimensional distributions are regularly varying with the same index. Before [13], limit theory for particular regularly varying stationary sequences was studied for the sample mean, maxima, sample autocovariance and autocorrelation functions of linear and bilinear processes with iid regularly varying noise and extreme value theory was considered for regularly varying ARCH processes and solutions to stochastic recurrence equation, see Rootzén [53], Davis and 1991 Mathematics Subject Classification. Primary 60F10, 60G70, secondary 60F05. Key words and phrases. Large deviation principle, regularly varying processes, central limit theorem, ruin probabilities, GARCH.