Entry requirements

120 credits including 90 credits in Mathematics. Integration Theory and Probability Theory II. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of the course, the student should be able to:

• construct probability measures on finite dimensional Euclidean spaces;
• use measure-theoretic and analytical techniques for the proof of limit theorems in probability;
• effectively use conditional expectations and Radon-Nikodym derivatives;
• outline proofs of important theorems of the course, and explain the main ideas of the proofs;
• characterise random variables with values in Euclidean and function spaces;
• use symmetry and invariance arguments;
• construct and compute with normal random variables in Euclidean spaces by cogently using basic notions of linear algebra;
• give examples of applications of probability, construct simple stochastic models and solve stochastic equations;
• decelop basic understanding of the fundamental random variables of probability theory, such as Brownian motion and the Poisson process.

Content

Quick review of basic probability. Probability spaces. Coin tosses and measure-theoretic foundations. Integration. Inequalities. Radon-Nikodym theorem and conditional probability. Convergence. Fundamental theorems of probability. Martingales. Characteristic functions. Gaussian spaces. Brownian motion. Point processes and Poisson. Other classes of function-valued random variables. Applications and additional topics.

Instruction

Lectures and problem solving sessions.

Assessment

Assignments given during combined with a home exam at the end of the course according to instructions given at the start of the course.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.