MATH 463 – Advanced Probability

MATH 463 provides a comprehensive treatment of the most common tools in Probability Theory, using a measure theoretic approach. The first part of the course presents the measure theoretic foundations of Probability, the formal definition of random variable as a measurable function and expectation as an integral in a measure space. From there, more advanced notions are presented, namely: the different types of convergence and their relationship (convergence a.s., in probability, mean-square convergence, weak convergence), characteristic functions and conditional expectations. The course illustrates these notions in a rigorous treatment of the main limit theorems: Laws of Large Numbers and Central Limit Theorem.


The course doesn't have formal prerequisites. Knowledge of basic Calculus-based probability is necessary (MATH 309). It helps having had a course on Measure Theory (MATH 401)

Why is it interesting for Probabilistic Modelers?

Most often, students and researchers active in Probabilistic Modelling have followed a classical Engineering curriculum providing them with a good knowledge of Probability, but without the measure theoretical approach. When it comes to carefully understand and work on advanced problems with random variables which are neither discrete nor continuous, but a mixture of both, or when it comes to understand the minimal assumptions required for limit theorems to hold, a measure-theoretic approach to probability is necessary.

The course will provide a rigorous coverage towards this objective. The course addresses mathematicians, as well as science and engineering students with a solid mathematics background. The concepts are rigorously proved as well as illustrated with important examples of applications. When one aims at studying stochastic processes, in particular in the continuous-time setting, a deep understanding of the most important Probability tools with a measure-theoretic approach is absolutely necessary.


Offered semi-annually in the Spring semester (usually on even years). Schedule varies.