This course includes two different parts each lasting two month. Both lectures will have a second part (continuation) in next summerterm.
Part 1: The Malliavin-Stein Method I: Poisson Processes (Oktober-November 2018)
This lecture is a special class for the lecture series of the Research Training Group RTG 2131 "High-Dimensional Phenomena in Probability - Fluctuations and Discontinuity", but it is also very appropriate for students in the master program. Stein's method is a collection of probabilistic techniques that allow to assess the distance between two probability distributions by means of differential operators. It has been discovered in the last decade that one can successfully combine Stein's method with the Malliavin calculus of variations. This so-called Malliavin-Stein method has become a versatile tool in many branches of probability theory and statistics. In our Tandem Lecture we present the foundations of Stein's method and that of the Malliavin calculus on the Wiener and the Poisson space. We then show how these techniques can be combined and how the resulting abstract error bounds evaluate in concrete situations. The applications we present are quantitative limit theorems for general functionals of Gaussian random fields and functionals that arise in stochastic geometry. During the winter term we concentrate on Poisson processes. The lecture will be continued in the summer term with Gaussian processes.
The lecture is held by Peter Eichelsbacher and Christoph Thäle from Ruhr University Bochum.
Part 2: Extremes for heavy-tailed time series (December 2018-January 2019)
This course aims at an introduction to extreme value theory for time series, i.e., for univariate or multivariate serially dependent sequences. Special emphasis will be given to time series whose marginal and finite-dimensional distributions exhibit power-law tails. For this reason, the notions of regularly varying random vector and regularly varying time series will be introduced and studied in detail. We will consider various time series models that have the regular variation property, including linear processes with regularly varying noise, GARCH processes, solutions to affine stochastic recurrence equations, stochastic volatility models, max-stable processes with Fréchet marginals. We will consider point process convergence for the suitably normalized time series and explain the crucial differences between the independent and dependent cases. In the former case, simple Poisson processes appear as weak limits while in the dependent case processes of compound Poisson type occur. We will introduce the necessary tools for point processes and their weak convergence on the way. We will also touch on large deviation theory based on regularly varying time series. It is closely related to the
extreme value theory for such sequences.
The course will be supported by lecture notes written by T. Mikosch andO. Wintenberger. This lecture is held by Thomas Mikosch, Kopenhagen.
You get 6 ECTS, Prüfungsform: An oral exam at the end of the semester.
For students of University Duisburg Essen this lecture allows to count within the Vertiefungsbereich Stochastik as well in "Special topics in Stochastic Processes" as in "Special topics in Stochastic Analysis".
Students from RUB, please contact the lecturers.