University of Duisburg-Essen

Faculty of mathematics, Campus Essen

Martin Hutzenthaler (martin (dot) hutzenthaler 'at' due (dot)de)

Anton Klimovsky (ak 'at' aklimovsky (dot) net)

8.September 2014

Lecture with exercises (4SWS+2SWS)
in the
Winter term 2014/15

Lecture: Tuesdays, 10.15-12.00 (room WSC-S-U-3.03) and Thursdays 10.15-12.00 (room WSC-S-U-3.03)

Exercises: Tuesdays 8.30-10.00 (room WSC-S-U-3.03)

Lecture: Tuesdays, 10.15-12.00 (room WSC-S-U-3.03) and Thursdays 10.15-12.00 (room WSC-S-U-3.03)

Exercises: Tuesdays 8.30-10.00 (room WSC-S-U-3.03)

**Probability theory II **

*Target audience:*

*Begin:*

*Content:*

- Martingale theory Optional sampling; optional stopping; Doob's inequality; martingale convergence theorem; backward martingales; de Finetti's theorem
- Convergence of measures Weak and vague convergence; Prohorov's theorem
- Product measures Theorem of Ionescu-Tulcea; consistent families; Kolmogorov's extension theorem
- Central limit theorem Theorem of Stone-Weierstrass; characteristic functions; the central limit theorem
- Brownian motion Stochastic processes; Gauss processes; construction and path properties of Brownian motion; strong Markov property and reflection principle; law of the iterated logarithm; Skorohod's embedding theorem
- Poisson point processes

*Requirements:*

Students are required to be familiar with measure theory and conditional expectations,
e.g., through the lecture 'Wahrscheinlichkeitstheorie 1' or through
chapters 1-8 in Klenke's book

*Exercises:*

Anton Klimovsky supervises the exercises.
You will find all exercise sheets on this
web page.

*Moodle course:*

There is a
moodle course on which everyone should
register. News and messages will be sent over this moodle platform.

*Language:*

The language both in the lecture and in the exercises will be English
or - if preferred - German.

*Literature:*

- Achim Klenke; Wahrscheinlichkeitstheorie, Springer, 2006
- Achim Klenke; Probability theory, Springer, 2008, translation of the book from 2006