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 Rootz\'en [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
