At the end of the course students will know the main results and principles of the Kolmogorov theory of probability, with a view to its application to the study of stochastic processes. Some applications of the martingale theory will be presented.
Course Prerequisites
A basic course of introduction to probability and study of intermediate analysis with basic measure theory and basic complex analysis will provide helpful background.
Teaching Methods
Lectures on the theory and introduction to problem solving through exercises assigned at home and done in the classroom.
Assessment Methods
Oral examination on the theoretical part together with check of some problems similar to those developed in the classroom.
Texts
D. Williams: Probability with martingales. Cambridge University Press, 1991
S. Resnick, A Probability Path, Brikhauser, 1999.
P. Billingsley: Probability and measure. 3nd edition. Wiley series in Probability and Mathematical Statistics, 1986.
R. M. Dudley: Real Analysis and Probability, 2002.
Contents
1. - Probability spaces, independence, random variables according to the Kolmogorov theory
2.- Expectation, integral, basic inequalities
3.- Convergence in probability. Relations with a.s. convergence and Lp convergence
4.- Laws of large numbers.
6. - Characteristic function of a probability distribution (Fourier-Stieltjes transform)
7.- Weak convergence of probability laws. The central limit theorem.
8.- Conditional expectation and conditional probability
9.- Martingales with discrete time. Applications of the martingale theory
10. - Hints on Borel probability measures in infinite dimensions
Course Language
Italian
More information
Students in the categories identified by the project on innovative teaching will have the opportunity to hold receptions also electronically and by appointment at times to be agreed with the teacher, or view the teacher's lecture notes.
