Faculty of Science

Objectives

To introduce most of the objects required for the understanding of probability theory. The themes listed are treated from a stochastic analytical point of view. These themes will recur in all branches of probabilistic analysis.

Contents

Convergence of probability measures: Weak convergence of probability measures, Fourier analysis; Relative compactness and tightness of families of probability measures; Prohorov’s theorem; Hillinger distance, Helinger integrals, and the absolute continuity and singularity of measures; Contiguity and asymptotic separation of measures; Modes of convergence of random variables and their interrelationships: Weak convergence, convergence in probability; Almost sure convergence; 0-1 Laws; Borel Cantelli lemmas; The law of iterated togasithm; Speed of convergence in the central limit theorem (large deviations); Strict stationary random sequences: Strict and wide sense stationarity; Measure preserving transformations; Ergodicity and mixing; Ergodic theorems.