Minicourse: "Use of Counterexamples in Probability for Research: part II"

13.11.2015, 10:15-11:45  –  Golm, Haus 9, Raum 1.10
Forschungsseminar Wahrscheinlichkeitstheorie

Jordan Stoyanov (Newcastle/Ljubljana)

We are going to have an intensive discussion on known objects: random variables and stochastic processes. The goal is to look closely at basic properties and results. Any result is proved under specific conditions: necessary and sufficient, only sufficient or only necessary. We will see how asking "right" questions about the role of the conditions, we arrive at non-trivial research problems. For obvious reasons we use the term counterexamples. To construct a counterexample is usually a challenging task. Beside the research aspect, important is that the counterexamples help us to achieve a better and deeper understanding of Probability theory, especially in learning and teaching.

Among the topics to discuss are: independence/dependence, distributions, different kinds of convergence, weak and strong law of large numbers (LLN), central limit theorem (CLT), other limit theorems, characterizations of distributions, characterizations of specific classes of stochastic processes, martingales, and more.

The audience will be naturally involved in all discussions. It may happen that we believe in our good knowledge of some area, however in fact there is much more (risk) behind. A series of new and open questions, i.e., new research projects, will be explicitly formulated, commented and referenced. One of the sources is the new edition of the book "Counterexamples in Probability. 3rd ed.", Dover, New York, 2013. References will be given also to several joirnal papers or arXiv manuscripts. Suggestions from the audience will be very welcome.