Probability and stochastic processes

Aims: To provide some basic knowledge on Probability Theory and Stochastic Processes. The level of the course strongly depends on the students' skills on such topics. Special attention is paid to those arguments which are commonly used in statistical inference, such as conditioning and asymptotics.

Learning outcomes: At the end of the course students will be able to: deal with advanced tools that lie at the foundations of modern probabilistic applications in Statistics; gain a deep understanding of stochastic models that are used to describe complex structures occurring in real world applications; profitably attend graduate courses on advanced topics in Statistics. 

Final exam: The exam is oral and consists of an interview with the teachers. However, the student may be required to discuss (not necessarily to solve) some simple exercises. The latter are obvious versions of the exercises developed by the teachers during the course.

Course contents

Module 1: Probability (prof. S .Mulinacci) 

·       Probability spaces

·       Independence

·       Measurability and random variables

·       Probability measures on (the Borel sets of) R and Rn

·       Integration and moments

·       Conditional expectation and conditional probability

Module 2: Stochastic processes (prof. P. Rigo) 

·       Characteristic functions

·       Limit theorems 

·       Some basic theory of stochastic processes (paths, filtrations, finite dimensional distributions)

·       Martingales

·       Brownian motion

·       Markov processes

References

P. Billingsley "Probability and Measure", John Wiley