Extreme value theory for group actions on homogeneous spaces

• Main Author: Kirsebom, Maxim Solund
• Published: University of Bristol 2014
• Subjects: 519.2
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.664970

In this thesis we study extreme value theory for random walks as well as one-parameter actions on homogeneous spaces. In both cases we investigate the limiting distributions for the maximum of an observable evaluated along a trajectory of the system. In particular we are going to consider asymptotic distributions for closest distance returns to a given point· and tor maximal excursions to the cusp. For closest returns on the torus we establish an exact extreme value distribution while for other cases we obtain estimates on the extreme value distributions for sparse sequences. For random walks we also obtain logarithm laws for the maximum. Finally we look into the extreme value statistics of exceedances of high levels in these settings. For the closest returns we establish convergence to a Poisson process for the point process of exceedances. In other cases we obtain estimates on the limiting distribution of the k'th largest maximum for sparse sequences.