2020/2021

2020/2021

DM270

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

1st semester (01/10/2020 - 20/01/2021)

9

84 lesson hours

Italian

ORAL TEST

PRIOLA Enrico (titolare) - 9 ECTS

A basic course of introduction to probability and study of intermediate analysis and measure theory will provide helpful background

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

1. - Probability spaces, independence, random variables

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

9.- Martingales with discrete time. Applications of the martingale theory

10. - Hints on Borel probability measures in infinite dimensions

Lectures on the theory and introduction to problem solving through exercises done in the classroom.

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.

Oral examination on the theoretical part together with check of some problems similar to those developed in the classroom.