Deep analysis of the Kolmogorov theory of probability, with a view to its application to the study of the general theory of stochastic processes.

Lectures on the theory and introduction to problem solving through exercises assigned in the form of homework.

Oral examination together with check of some of the problems assigned as homework.

Study of intermediate analysis and measure theory will provide helpful background

1.- Kolmogorov probability space. Construction through the extension theorems of Kolmogorov and Ionescu-Tulcea.
Analysis of the condition of stochastic independence.
2.- Expectation, basic inequalities (Tchebyshev, Jensen maximal Kolmogorov) convergence of sequences of random elements: in probability and almost sure: Borel-Cantelli lemmata and other 0-1 laws (Kolmogorov, Hewitt-Savage).
3.- Integral transformations of probability distributions.
4.- Laws of large numbers: Khintchin weak law, Etemadi strong law.
5.- Weak convergence of probability laws: the Prokhorov theory. The central limit theorem: the Lindeberg formulation for triangular arrays of independent random numbers.
5.- Conditional expectation as Radon-Nikodym derivative and as projection (regression function). Existence of regular conditional distributions.

6.- Sequences of random numbers forming a (s)martingale: convergence, optional stopping theorems and applications to real analysis, maximal inequalities, gambler ruin problem, stong laws of large numbers.

In addition to teacher's notes, see: Erhan Cinlar (2011) Probability and Stochastics. Springer.