SITO NON PIU' AGGIORNATO - UNIVERSITÀ DI PAVIA

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### Probability

Professors:
Regazzini Eugenio, Rigo Pietro
Year:
2015/2016
Course code:
500691
ECTS:
9
SSD:
MAT/06
DM:
270/04
Lessons:
84
Period:
I semester
Language:
Italian

#### Objectives

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

#### Teaching methods

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

#### Examination

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

#### Prerequisites

Study of intermediate analysis and measure theory will provide helpful background

#### Syllabus

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.

#### Bibliography

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

#### Modules

Professor:
Regazzini Eugenio
Lessons:
28
ECTS:
3
SSD:
MAT/06

Professor:
Rigo Pietro
Lessons:
56
ECTS:
6
SSD:
MAT/06

Dipartimento di Matematica ''F. Casorati''

Università degli Studi di Pavia - Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602