Abstract:

In the last 20 years, much insight has been gained into the dependence structure of time series models beyond linear processes (ARMA, FARMA,...). A driving force was the rapid development of financial time series analysis. A large variety of new non-linear models was introduced, such as the GARCH and stochastic volatility models for return data. These models are heavy-tailed white noises which cannot be characterized in a satisfactory manner by autocorrelations; additional characteristics beyond moments are required. A second driving force of non-linear time series modeling was teletraffic where specific models were developed since the beginning of the 1990s to describe the deviations of teletraffic data from classical queuing model behavior. It was discovered early on by Taqqu, Willinger, and coworkers that components of the considered models have to have extremely heavy tails (such as file sizes and transmission durations of files from one source to another) if one wants to describe real-life teletraffic phenomena.

Returns, teletraffic data, insurance claims, perpetuities, etc., have in common that they are "heavy-tailed" in the sense that sufficiently high moments are infinite. In this course, we focus on time series models with heavy tails (in the sense that the finite-dimensional distributions have power law tails). We call such time series "regularly varying".

By now, the theory of regularly varying stationary sequences is well understood, starting from the paper Davis and Hsing (1995, Ann. Probab.). Already in this paper, some fundamental problems were solved: point process convergence for weakly dependent regularly varying sequences, weak convergence of extremes, infinite variance central limit theory, large deviations.

In this course, we show how the theory of non-linear regularly varying time series has developed in the last 15-20 years. We will consider the calculus of multivariate and sequential regular variation and its various applications to asymptotic theory for partial sums, maxima, order statistics, point processes, other structures. We will consider how these results can be used to derive specific extremal cluster indices, depending on the structure under consideration. We will also derive precise large deviation results and bounds for the ruin probabilities of random walks with dependent regularly varying steps. We will touch on some newer developments such as max-stable stationary processes (whose heavy-tail versions are regularly varying), following work by Taqqu, Stoev, Kabluchko, Schlather, de Haan, and others, and the conditional tail chain theory developed by Basrak and Segers (2009, SPA). The theory will be illustrated by suitable linear and non-linear time series. Among the latter ones are the GARCH and stochastic volatility models for returns. Particular attention will be given to perpetuities (or random affine mappings, stochastic recurrence equations) which have been intensively studied since Kesten (1973, Acta Math.) discovered that these processes are heavy-tailed.

The course aims at graduate students, PhD students and postdocs with interest in probability theory, statistics, applied probability, time series analysis, extreme value theory and dependence modeling.

The lectures will be supported by notes and extensive references to the relevant literature. The monographs by Sid Resnick "Heavy-tail phenomena" and "Extreme values, regular variation, and point processes" (Springer, 2007 and 1987) contain useful background material on univariate and multivariate regular variation and heavy-tailed applied models.

Time:            Tuesdays 10:15-12
Auditorium:  HG G 43 (HWZ)
Begins:         tba
 

M. Struwe