## Probability Theory (MSc course in mathematics)

Roster, Email alias, Lecture notes and assignments: available to those attending the course

Prerequisites: A standard course in probability theory (e.g., at the level of David Williams, Probability with Martingales, Cambridge Univ. Press, 1991), a course in mathematical analysis, some logic and set theory, ability to think independently.

Contents: Foundations of probability theory. Integration and differentiation of measures. Construction of probability measures. Infinite products. Independence. Coupling. The fundamental theorem of probability. Limit theorems. Sums of independent random variables. Martingales. Convergence of probability measures. Elements of ergodic theory. Symmetry and invariance. Introduction to some standard random variables, such as Gaussian random variables, Poisson process, Brownian motion, random trees, Lévy processes. Depending on time, we may discuss some applications in mathematical analysis, in information theory, in queueing theory and in physics.

Goals: Besides those stated in the university catalog, a course in probability should form the basis for other courses in the mathematics department and beyond, ranging from statistics to partial differential equations and modern applications of probability such as, e.g., in finance and in mathematical physics. Probability theory is one of the central topics in mathematics and a foundational course should be in the tool of tricks of everyone. My philosophy in teaching is not to be exhaustive (this is a vacuous statement) but rather explain some central concepts, some of the big ideas in probability, rigorously derive some of its basic theorems, and explain some of its uses.

Assessment:

• Take-home exam

Bibliography:

• Patrick Billingsley. Probability and Measure. Wiley, 1995, third edition.
• Leo Breiman. Probability. Addison-Wesley 1968 (latest edition by SIAM: 1992).
• Kai Lai Chung. A Course in Probability Theory. Academic Press, 1974 (latest edition 2000).
• Richard Durrett, Probability: Theory and Examples.
• William Feller. An Introduction to Probability Theory and its Applications. Wiley. Volume 1: 1957. Volume 2: 1966.
• Bert E. Fristedt and Lawrence F. Gray. A Modern Approach to Probability Theory.
• Boris V. Gnedenko. The Theory of Probability. Chelsea 1967. Translated from the 4th Russian edition.
• Olav Kallenberg. Foundations of Modern Probability , Springer-Verlag, latest edition: 2010.
• Davar Khoshnevisan, Probability, American Mathematical Society, 2007.
• Achim Klenke, Probability Theory: a Comprehensive Course, Springer-Verlag Universitext, 2007.
• Albert N. Shiryaev, Probability. Springer-Verlag. latest edition by Cambridge U. Press. Birkhäuser, 1996.
• Leonid Koralov and Yakov G. Sinai, Theory of Probability and Random Processes, Springer-Verlag Universitext, 2007.
• David Pollard, A User's Guide to Measure Theoretic Probability, A User's Guide to Measure Theoretic Probability. Cambridge Univ. Press, 2001.
• Daniel W. Stroock, Probability Theory: An Analytic View, Cambridge Univ. Press, 2010.