Copulas and spectral measures: Statistical modelling under long-range dependence

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Many observed phenomena cannot be explained by linear dependence structures. Moreover,often observed time series exhibit either linear or nonlinear long-range dependence. A general approach to modelling dependence is provided by copulas. In spite of an extended literature, current copula models are often not easy to interpret and at the same time fail to capture the dependence structure of extremes. Moreover, at this stage, no results on copulas in the context of long-range dependence seem to be available. These issues are addressed in this project. The project is divided into three main parts. In Part I, applied aspects of statistical modelling with copulas and extremal copulas are explored under independence or weak dependence. We focus on finding models capturing a wide range of dependence structures for multivariate extremes. In Parts II and III statistical inference for (extreme value) copulas and other model parameters is addressed in the context of linear (Part II) and nonlinear (Part III) long-memory processes with finite or infinite second moments. Estimation, testing (goodness-of-fit, linearity) and conditional risk measures will be considered. Due to the copula structure and long-range dependence, nonstandard limit theorems are expected to emerge in Parts II and III.