SMI Perugia 2023

Course "Probability and Markov Chains"

Prof. Wolfgang Woess



All interested particpants are asked to contact W. Woess well before the beginning of the course by email at woess@tugraz.at: please specify which are the topics listed below in "Programme" with which you are already familiar from your studies. You may also express preferences.

This website provides access to the course material.

Please note that for downloading the pdf files you will need
Username: family name of the person who teaches the course
Password: the capital of Umbria
(first letters uppercase)

Programme

The programme will consist of a part of the following sub-topics. The choices will depend on the preliminary knowledge and preferences of the participants.

  1. Quick reminder: probability spaces, random variables, expectation, typical probability distributions.

  2. Convergence of random variables.

  3. Law of Large Numbers. Variant A: classical (Kolomogorov) Variant B (more advanced): via Birkhoff's ergodic theorem (the latter to be proved).

  4. Characteristic functions (= Fourier transformation) and Central Limit Theorem.

  5. Martingales, exchangeability.

  6. Basics of Markov chain theory.

  7. Topics from advanced Markov chain theory, links with potential theory.


Knowledge of measure theory is indipsensable. It is suggested that participants who are not familiar with it undertake some preliminary reading. Particularly suitable: the initial parts of the books by Durrett and Klenke, as well as the notes "Basic Measure Theory". This material can be found online at the links provided below.

Recommended literature


  1. Rick Durrett: Probability Theory and Examples. Cambridge University Press,first edition 1990, fifth edition 2019. pdf

  2. Achim Klenke: Probability Theory - A Comprehensive Course. Springer Universitext, first English edition 2008, third English edition 2022. pdf

  3. Wolfgang Woess: Denumerable Markov Chains. EMS Textbooks in Mathematics, European Math Soc., 2009. pdf

  4. Heinrich v. Weizsäcker: Basic Measure Theory. Technische Universität Kaiserslautern 2004/2008. pdf

Exercises

Participants should autonomously solve exercises and present them at the blackboard for discussion.

FIRST EXERCISES

SECOND SHEET OF EXERCISES

THIRD SHEET OF EXERCISES

FOURTH SHEET OF EXERCISES

FIFTH SHEET OF EXERCISES

(Same password and username as above!)

Final evaluation

Based on presentations of exercises and the final exam.

Here is the text of the exam.

And here are the solutions.

Pictures

You can download a zip-file with some photographs HERE As well as a second one HERE (Same password and username as above!)



To the website of the Scuola Matematica Interuniversitaria in Umbria's capital Perugia



To the homepage of Wolfgang Woess



Last modified on August 9, 2023.

Impressum:
Für den Inhalt verantwortlich: Wolfgang Woess
Institut für Diskrete Mathematik
TU Graz, Steyrergasse 30, A-8010 Graz