Multivariate stochastische Integrale mit Anwendung am Beispiel Operator-stabiler und Operator-selbstähnlicher Zufallsfelder

Roughly speaking, the content of this thesis can be divided into two parts. On the one hand, we will deal with multivariate random fields in form of {X(t) : t is element of R^d}, where X(t) is an R^m-valued random vector for each t of R^d, and then we will investigate the existence of such fields which are operator-self-similar in the sense of [29] and whose margins, at the same time, have a real operator-stable distribution. This will be done in terms of a harmonizable and a moving-average integral representation with respect to an appropriate independently scattered random measure, respectively. Hence, we actually extend the results in [40], [5] and [29], whereas the aspect of operator-stability is mentionable and seems to be more natural in the context of multivariate questions. Additionally, we will be able to present some further properties. In particular, a special case of the harmonizable representation permits us to construct continuous modifications and can therefore - among other examples - be regarded as a source for subsequent researches.
On the other hand, we have to ensure that the applied random measures already provide the property of operator-stability which is intended for the resulting margin distributions of the random field. So we have to cope with the challenges that occur concerning the related stochastic integrals as this will be the method for constructing these random fields. This problem will be solved in a very comprehensive way by unifying and extending the results in [35] for the univariate case towards the multivariate one via vector valued measures and even by adding a complex-valued perception as proposed in [40]. In this context we will also be able to show that these random measures always have to be infinitely divisible as long as they are atomless. In view of their universality our results appear rather self-contained and could also serve as a useful tool for extensive modeling problems in the future.
Furthermore, we state a potential definition for the domain of attraction of multivariate random fields which looks quite general, but which still allows a nice characterization for the operator-self-similarity (of a given random field) by extending the corresponding univariate result in [17].
In spite of the mentioned flexibility it will turn out that the consideration of linear operators and generalized polar coordinates (see [5]) is some kind of central theme throughout the whole thesis which finally leads to interesting and practicable examples.