### Advanced Probability Theory

Monday 11:30-13:00, Seminarraum 10 (OMP 1)

Wednesday 11:30-13:00, Seminarraum 08 (OMP 1)

**Proseminar** (optional): Monday 13:15-14:00,
Seminarraum 10 (OMP 1)

**Summary:** The lecture presents the most important
results and concepts of the modern probability theory, using
the measure-theoretic framework, in the context of infinite
sequences of random variables, i.e. stochastic processes in
discrete time. It gives an introduction to basic convergence
results for such sequences: laws of large numbers, central
limit theorem, large deviation for independent sequences;
convergence of stochastic series; martingale convergence
theorems; convergence of Markov chains. We discuss weak
convergence of probability measures on function spaces and
construct the Brownian motion by means of Donsker's
theorem.

**Literature:**

- R. Durrett: Probability theory & examples. (Duxbury Press, 1996)
- A. Klenke: Wahrscheinlichkeitstheorie. (Springer, 2008)
- D. Williams: Probability with martingales. (Cambridge University Press, 1991)
- P. Billingsley: Probability and measure. (Wiley, 1986)
- A. Dembo: Probability Theory (lecture notes, Stanford University, download)

**Lecture notes:**

- Lecture notes [PDF], (new version from June 1, covers essentially all the material discussed in the lecture, but the final chapters need a lot of cleaning)
- Handwritten lecture notes for the same lecture given in SS2012 [PDF]

**Questions for the proseminar:**

- Serie 1 (March 14) [PDF]
- Serie 2 (April 4) [PDF]
- Serie 3 (April 11) [PDF]
- Serie 4 (April 18) [PDF]
- Serie 5 (April 25) [PDF]
- Serie 6 (May 2) [PDF]
- Serie 7 (May 9) [PDF]
- Serie 8 (May 23) [PDF]
- Serie 9 (May 30) [PDF]
- Serie 10 (June 13) [PDF]

### Seminar in Probability Theory

Wednesday 9:45-11:15, Seminarraum 08 (OMP 1)

First session: 2 Mar 2015

The topics of the seminar will be 'TBD'.

Detailed list of topics will be given on March 2

**Literature:**

**Programm:**

### Übungen zu Wahrscheinlichkeitstheorie und Statistik

Wednesday 15:00-16:30, Hörsaal 2 (OMP 1)

Fängt am 9. Mars an

Weitere Informationen zu der Vorlesung und Übungsbetrieb: Seite der Vorlesung