Probability theory

Postgraduate course

Course description

Objectives and Content

The course provides an introduction to measure theoretic probability theory. Among the topics discussed are: convergence of random variables, classical limit theorems such as the law of large numbers and the central limit theorem, conditional expectation and martingales. Applications of the theory will be discussed e.g., to statistical inference theory and mathematical finance.

Learning Outcomes

After completed course, the students are expected to be able to:

  • Know and understand basic concepts within measure theoretic probability theory, including sigma-algebra, probability measure, random variable, independence, and conditional expectation.
  • Understand convergence in probability, almost sure convergence, and convergence in distribution, as well as the relations between these convergence concepts.
  • Apply common methods such as the Borel-Cantelli and Kolmogorovs 0-1-law to treat and describe the convergence of sequences of random variables.
  • Reproduce the key ideas that form the basis for the proofs of the most central limit theorems in probability theory, including the law of large numbers and the central limit theorem.
  • Apply the most important limit theorems and Slutsky┬┐s theorem for asymptotic analysis in statistical inference theory
  • Know and be able to use concepts such as filtration, conditional expectation, and martingales in discrete time.
  • Apply the theory of martingales to simple models from mathematical finance
Compulsory Assignments and Attendance
Exercises. Valid for 2 semesters.