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Probability Theory III

This course covers basic measure theory, convergence and limit theorems, and martingale theory. This material and the techniques learnt during this course have important theoretical and applied uses within stochastic methods.

In this course we lay out the rigorous foundation for probability theory, based on measure theory. We will see how this will provide us with the tools for dealing with expectation of random variables in a general way, and thus avoid treating the discrete and continuous cases separately. We move on and revisit the fundamental topic of convergence in probability theory, and go through the important theory for sequences of random variables known as martingales. The course will in roughly equal proportions treat the topics:

  • Basics of measure theory;
  • Convergence and limit theorems;
  • Martingale theory.