University of Duisburg-Essen
Faculty of mathematics, Campus Essen
(martin (dot) hutzenthaler 'at' due (dot)de)
Anton Klimovsky (ak 'at' aklimovsky (dot) net)
Lecture with exercises (4SWS+2SWS)
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)
Probability theory II
This lecture is both for Masters students and Bachelor students.
This lecture is an 'Aufbaumodul Stochastik' (9ECTS).
The work load is about 270 hours (90 hours of presence).
The first lecture will be on Tuesday, October 14, 2014.
The exercises start on Tuesday, October 14, 2014, with a short repetition.
- 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
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
Anton Klimovsky supervises the exercises.
You will find all exercise sheets on this
There is a
moodle course on which everyone should
register. News and messages will be sent over this moodle platform.
The language both in the lecture and in the exercises will be English
or - if preferred - German.
- Achim Klenke; Wahrscheinlichkeitstheorie, Springer, 2006
- Achim Klenke; Probability theory, Springer, 2008, translation of the book from 2006