Stochastics

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, G. Dierickx (Fwo) and D. Thumas are Ph.d. students. 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.