2019/2020

2019/2020

DM270

MAT/06 (PROBABILITY AND MATHEMATICAL STATISTICS)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

1st semester (30/09/2019 - 10/01/2020)

9

84 lesson hours

Italian

ORAL TEST

RIGO PIETRO (titolare) - 9 ECTS

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. 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.- Transformations of probability distributions.

4.- Laws of large numbers: Khintchin weak law, Etemadi strong law.

5.- Weak convergence of probability laws. 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 and/or proper 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.

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

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

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