This thesis concerns continuous-time Lévy-driven moving averages with deterministic kernels. These are motivated as a noisy component of a more general model using intuitive concepts and then deduced and specified by more rigorous mathematical theorems. The driver will use a heavy-tailed distribution chiefly to accommodate certain real-world phenomena such as rare events being non-negligible in terms of a probabilistic analysis. The endeavour is to understand the possible dynamics of such driven moving averages—indeed, this is key to extracting or inferring central aspects of the chosen model from actual data. In more concrete terms, using, extending and proving novel limit theory for variational statistics, e.g. the power variation, of given stationary data in this heavy world we shall develop a statistical methodology which allows inference and in particular estimation in a large class of parametrized Lévy-driven moving averages for which the dynamics are allowed to be surprisingly preposterous. Even more concretely, we shall, among other tasks, derive first- and second-order asymptotics for an estimator which finds the optimal parametric model by comparing, under a suitable weighing, the variational statistic of the empirical distribution with the theoretical counterpart of a possible distribution provided by the mode—a procedure aptly named the minimal contrast method.
Thesis advisor: Mark Podolskij