## Organisation

# Applied Mathematics

### Research Group

This research unit has two permanent members: U. Einmahl (Probability and Mathematical Statistics) and T.Kadankova (Stochastic processes and their applications). The other two members, J. Dony (postdoc) and A. Van Messem (Ph. D. Student), do research in mathematical and applied statistics. The research by Uwe Einmahl focuses on limit theorems of probability in general spaces. Classical probability theory deals mainly with random variables taking values in the real line is well developed, but there are many open questions for random variables taking values in infinite-dimensional spaces. For instance, there is no natural extension of the classical central limit theorem to Banach spaces. Also the so-called law of the iterated logarithm which in the classical setting was established in 1942 has only relatively recently (1986) been obtained for infinite-dimensional spaces. The most recent research by U. Einmahl has addressed not only theoretical questions such as refinements of the basic limit theorems, but also applications to statistics such as density estimation which are possible via so-called empirical processes since these processes can be considered as random elements in a suitable infinite-dimensional space. Tetyana Kadankova's research is concerned with Lévy processes, especially with so-called one- and two-sided exit problems for such processes. A problem which she has recently investigated is determining the laws of the first passage of a level (the first exit time from a fixed interval) by such processes. Lévy processes are considered interesting objects both for the theory and applications. For this reason, this class of stochastic processes has received much attention during the last years. Some important applications of this topic come from financial mathematics and insurance. Oscillating Lévy processes serve also as governing processes for oscillating queueing systems and thus they are also important in queueing theory. Another part of her research is devoted to semi-Markov random walks and compound renewal processes. Additionally, she studies stochastic processes reflected at their infimum (supremum) which serve as governing processes in various applications. Julia Dony has an FWO postdoc position (2008-2011) and she works on applications of empirical process theory to nonparametric statistics. This is the continuation of her Ph.D. Thesis (under the guidance of U. Einmahl and D. Mason, co-promotor) which she defended in May 2008 at the VUB. Arnout van Messem currently works on a Ph. D. Thesis (under the guidance of U. Einmahl and A. Christmann, co-promotor) which he will defend during the academic year 2010/11. The main topic of this thesis are so-called support vector machines which are important objects in " robust" statistics.