250100 VO Advanced probability theory (2016S)

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
It further discusses the theory of the weak convergence of probability measures on function spaces, aiming on a rigorous construction of the Brownian motion via Donsker's theorem.