Probability theory is the branch of mathematics concerned with the analysis of random phenomena. The members of the ISM Probability Group are involved in research in a broad range of areas spanning theoretical and applied, continuous and discrete probability. A particular focus is on the development and analysis of probabilistic models for real-world phenomena from physics, biology, statistics and computer science. Some specific topics of interest are: statistical physics in a random environment, branching systems in biology, distances and random energy landscapes, data structure analysis using random trees, genetics and population biology.
Many members of the group are also members of the CRM Probability Lab.
Students interested in graduate study in any of the areas cited above are invited to apply for admission to the program. There are no formal prerequisites other than those required by the departments. The following guidelines should be followed, however, and courses selected in consultation with an advisor from the group.
Students in the program are expected to have mastered the subject matter of the undergraduate curriculum in probability theory. All students are required to take the basic courses: Real Analysis Measure Theory and Probability Theory. Students are then expected to take a number of more specialized courses.
Probability spaces. Random variables and their expectations. Convergence of random variables in Lp. Independence and conditional expectation. Introduction to Martingales. Limit theorems including Kolmogorov's Strong Law of Large Numbers.
Characteristic functions: Elementary properties, Inversion formula, Uniqueness, convolution and continuity theorems.
Weak convergence: Portmanteau theorem, Sequential compactness, tightness, Prohorov's theorem, Polish spaces, Central limit theorem, Skorokhod's representation theorem.
Stochastic processes: General theory. Kolmogorov Extension theorem, Kolmogorov continuity theorem. Regular probability spaces and conditional distributions, probability kernels. Construction of Brownian motion, Donsker's theorem
Exchangeability: De Finetti's theorem, The Aldous-Hoover theorem.
Other topics if time permits.
This is an introductory course on Lévy processes with an emphasis on fluctuation theory for spectrally negative Lévy processes (SNLPs), including compound Poisson with drift and Brownian motion with drift. Applications in ruin theory, operations research and/or financial mathematics will be discussed, in view of students interest.
Schedule: Tuesday and Thursday, 2-3:30pm (could be modified at the first lecture)
The theory of large deviations studies probabilities of events which, in a large sample, are exponentially rare in the number of samples. We will cover standard methods including large deviations for i.i.d. sequences, correlated variables, and occupation measures. This theory has found many applications in finance and risk management, simulation and sampling, as well as operations research and statistical mechanics. The main part of the course will cover: Introduction to large deviations, the large deviations principle, Sanov’s theorem and method of types, Cramer’s theorem, Gartner-Ellis theorem, concentration inequalities, large deviations for Markov chains. We will also discuss the application of the theory to constructing efficient sampling mechanisms, such as importance sampling. The prerequisites for the course are basic probability theory and stochastic processes.