240111 Probability Theory (V) (WiSe 2012/2013)

This course introduces the students to the fundamentals of rigorous probability theory on the basis of measure theory.

The first half is devoted to an introduction to measure theory. After a brief introduction on how to formulate the concept of probability rigorously, the students will learn the concepts of sigma-algebras and measures, integration with respect to measures and fundamental convergence theorems. Then the problem of existence and uniqueness of measures is discussed and some basic measures on the Euclidean spaces are constructed. Fubini's theorem on integration on product spaces, another fundamental theorem in measure theory, will be also presented. By utilizing measure theory introduced so far, the notions of probability and random variables are formulated rigorously and some important concepts in probability theory such as independence of random variables are also introduced. Kolmogorov's extension theorem on existence of sequences of random variables is explained and various examples of (both discrete and continuous) random variables are also presented.

The latter half mainly concerns various limit theorems for random variables. The earlier part will treat laws of large numbers, several types of convergence of random variables and convergence of sums of independent random variables.
The remaining part of the course is devoted to description of convergence of probability laws on Euclidean spaces and aims to state and prove the central limit theorem, in relation to which properties of characteristic functions and of normal
(Gaussian) random variables are also discussed.

The intention of the whole course would be to prepare the students to go on to advanced topics in probability theory such as martingales, Brownian motion and Ito's stochastic calculus.

Requirements for participation, required level

The participants are supposed to be familiar with:
Logic and Sets
Multivariable Calculus

Bibliography

Jacod, J. and Protter, P., Probability Essentials, 2nd edition (corrected 2nd printing), Springer, 2004. [Chapters 1-21]
Rudin, W., Real and Complex Analysis, 3rd edition, McGraw-Hill, 1987. [Chapters 1, 3 and 8]
Dudley, R. M., Real Analysis and Probability, Cambridge University Press, 2002. [Chapters 3, 4, 8 and 9]

External comments page

http://www.math.uni-bielefeld.de/~nkajino/lectures/Prob2012.html