Über Zusammenhänge von leichten Tails, regulärer Variation und Extremwerttheorie

Abstract

This work deals with some aspects of the extremal behavior of both light tailed and heavy tailed distributions. The thesis is divided into three parts and starts with the analysis of limit laws for $l_p$-norms of positive i.i.d. random vectors which establish a connection between limit laws for sums of i.i.d.\ random variables and extreme value theory. Here, a new approach allows us to unify the analysis for all three max-domains of attraction, where special emphasis is laid on the Gumbel case. The second part of the thesis deals with the extremal behavior of certain time series which bear resemblance to so-called ``random difference equations'' (RDEs). We analyse the behavior of an underlying and an observable time series, given an extreme event in the observable one, by extending results for a single time series to the case of two connected time series. In the third part of the thesis we take a closer look at the heavy tailed behavior of RDEs. In order to derive ! the characteristic $\kappa$, the index of regular variation, we propose a new method based on the results of Kesten (1973).