Probability Theory

A.Y. 2017/2018
9
Max ECTS
68
Overall hours
SSD
MAT/06
Language
Italian
Learning objectives
Probability theory is now applied in a variety of fields including physics, engineering, biology, economics, social sciences, ... This course is an introduction to the rigorous theory of probability. The perspective theme is the Doob's theory of discrete time martingales. The Kolmogorov strong law of large numbers and the theorem of three series are proved with martingale techniques. In addition, the central limit theorem is proved together with the main results on week convergence and characteristic functions. In the part of exercises, the first results of Markov chains are introduced.
Expected learning outcomes
Knowledge of the topics of the course and their application to theoretical problems.
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
Course syllabus
Part I. Foundations:
Measure spaces
Events
Random variables
Independence
Integration
Expectation
Strong law of large numbers
Product measure
Gaussian vectors

Part II. Martingale Theory:
Conditional expectation
Martingales
The convergence theorem
UI Maringales, L1 convergence and applications
L2 Maringales, angle-brackets process and relation with martingale' convergence

Part III. Compendia of theory
Markov chains
Weak convergence. Tightness. Lévy's Convergence Theorem.
Central Limit Theorem
MAT/06 - PROBABILITY AND STATISTICS - University credits: 9
Practicals: 20 hours
Guided problem-solving: 6 hours
Lessons: 42 hours
Professor(s)
Reception:
on appointment
office 2099
Reception:
Please write an email
Room of the teacher or online room