Probability Theory

We start by embedding probability theory into a general theory of measure and integration. This will allow us to derive theorems that may not have been included in the Analysis III course but that are important for construction probability theory axiomatically. Examples are theorems on the existence of infinite sequences of independent random variables or the theorem of Radon and Nikodym.

Once we have such sequences of random variables we will derive 0-1-laws and, as a consequence, a Strong Law of Large Numbers.

To analyze fluctuations we will present a Fourier-theoretic approach to the Central Limit Theorem. In order to be able to study more general sequences of random variables we will introduce the concept of conditional expectation. In the end we will give a brief introduction into martingales as a concept of rather general stochastic processes.