93762 - Advanced Probability

Course Unit Page

Academic Year 2021/2022

Learning outcomes

By the end of the course the student acquires the fundamental notions of the theories of conditional probability and weak convergence of probability measures. In particular, the student is able to investigate the asymptotic behaviour of sequences of probability measures arising in statistical inference and to coherently employ the concept of conditional probability in the statistical investigation.

Course contents

Probability spaces

A few elementary notions of measure theory

Random variables and their probability distributions

Probability measures on R and R^n

Moments

Conditional distributions

Characteristic functions

Weak convergence of probability measures

Other modes of convergence

Laws of large numbers

Central limit theorems

Readings/Bibliography

Billingsley P. (1986) Probability and measure, Wiley, New York

Dall'Aglio G. (1987) Calcolo delle probabilita', Zanichelli.

Grimmett G. and Stirzaker D. (2001) Probability and random processes, Oxford University Press.

Teaching methods

Lectures and class exercises

Assessment methods

The first part of the exam consists of a written assignment. Usually, the latter consists in 4 or 5 simple exercises which are obvious versions of those solved by the teacher.

The second part of the exam, subject to overcoming the first, lies in an oral interview. The possible questions may concern each part of the course. Typically, the interview starts with a very general question (such as "Conditional distributions" or "The central limit theorem") and then, as the topic is introduced, they become more specific. In addition to knowledge of the topics discussed in the course, evaluation criteria are the skill to connect different arguments and the adequacy and consistency of the adopted language. A mnemonic exposition, as well as the inability to discuss with the teacher, are penalized. In other terms, it is important to be able to discuss with the teacher, to be interrupted, and possibly to address some simple objections.

The above remarks do not depend on whether the exam is online or in presence. However, for online interviews, it is desirable (even if not mandatory) that the camera is able to frame the sheet where the student is writing.

Teaching tools

Notes and the text-books quoted above

Office hours

See the website of Pietro Rigo