Extremal Behavior of Time Series: Refined Models, Analysis and Inference

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The aim of this project is the development of new probabilistic and statistical methods which allow for a refined description of the extremal behavior of asymptotically independent time series, i.e. time series which show no clustering of extreme values in the limit. So far, a well explored theory about time series extremes exists only in the asymptotically dependent case. However, the “classical” asymptotic approach of extreme value theory neglects the fact that a variety of different behaviors may evolve in the “pre-asymptotic” behavior of a time series. For example, many asymptotically independent time series models show a decent amount of clustering of large values in finite samples sizes. By a combination of models for asymptotically independent random vectors and methods of asymptotically dependent time series, we aim at a refinement of the tools of extreme value theory that allow for a better description and distinction of different types of asymptotic independence. We tackle both the probabilistic aspects by the development of new limit processes for cases in which classic theory gives only degenerate results, and the statistical side by giving estimators for different aspects of extremal dependence and statistical procedures for model estimation and validation. A prominent example of asymptotically independent time series is the well-known class of stochastic volatility models. Therefore, a special focus is laid on the analysis of financial time series, especially with regard to the suitability of this class for modeling the extremes of financial data.