58996 - Probability

Learning outcomes

At the end of the course the student: - has acquired mathematical bases of Probability Theory at an advanced level and some important results on sequences of independent variables, stationary sequences, convergence of probability measures on metrizable spaces, Fourier transform of probability measures, martingale theory with discrete time; - he is able to apply acquired knowledge to the study of stochastic processes.

Course contents

Probability spaces. Probability measures. Extension theorem. Events, random variables. Expectation and integral. Stochastic independence. Stationary sequences of random variables. Probability on metrizable spaces, weak convergence. Fourier transform of probability measures. Martingales in discrete time.

Readings/Bibliography

N. Pintacuda. Probabilità. Zanichelli.

P. Billingsley. Probability and measure. Wiley.

Teaching methods

The teaching of the course is  based on lectures aimed at providing students with the bases for the study of Probability Theory at an advanced level, in particular abstract measure theory and martingale theory, The lectures will be directed at establishing relations with other areas of mathematics such as analysis,  topology and the theory of dynamical systems  and will be supported by examples and exercises.

Assessment methods

Final verification consists in an oral test.

The oral tests is based on three questions. In answering them the student will have to show that he/she masters the basic concepts of the course and that he/she is able to develop rigorous arguments and to solve simple exercises on the content of the course.

Teaching tools

Lectures.

Office hours

See the website of Massimo Campanino