Forelesninger/lessons

In our last (home) lessons (31. March and 2., 7. April) we studied different statistical methods of extreme value distributions, that is the block maxima method for both stationary and non-stationary data and the approach of Pickands-Balkema-de Haan (see Ch. 4 in the lecture notes or Ch. 6 in Embrechts).

This week we started with the discussion of the weak convergence of point processes which can be used to study extremal events from  a more general and deeper point of view (see Ch. 5 in the lecture notes or in the book of Embrechts). The point process of exceedances, which counts the number of claim sizes over a certain threshold, is an example of a point process. The weak convergence of this process implies e.g. the distributional convergence of (normalized) maxima or order statistics of i.i.d. claim size variables.

Our goal in the coming lessons in April/May is to gain a solid understanding of the basic results of the theory of point processes. The latter pertains e.g. to the construction of Poisson random measures (as a tool for the construction of Levy processes in connection with applications in finance), marked point processes, the characterization of weak convergence of point processes by means of Laplace functionals, Poisson approximation, Kallenberg’s theorem as a criterion for weak convergence of simple point processes and the link between the theory of point processes and extreme value theory.

As mentioned earlier Ch.1-4 in the lecture notes form the pensum for the exam.