# NMAK11003U Advanced Probability Theory 1 (VidSand1)

Volume 2022/2023
Education

MSc Programme in Actuarial Mathematics
MSc Programme in Mathematics
MSc Programme in Mathematics-Economics
MSc Programme in Statistics
MSc Programme in Mathematics with a minor subject

Content
• Sequences of random variables, almost sure convergence, Kolmogorov's 0-1 law.
• The strong law of large numbers.
• Weak convergence of probability measures. Characteristic functions.
• The central limit theorem. Triangular arrays and Lindebergs condition. The multivariate central limit theorem.
• The ergodic theorem.
Learning Outcome

Knowledge:

• Fundamental convergence concepts and results in probability theory.

Skills: Ability to

• use the results obtained in the course to verify almost sure convergence or convergence in law of a sequence of random variables.
• verify conditions for the central limit theorem to hold.
• translate between sequences of random variables and iterative compositions of maps.

Competences: Ability to

• formulate and prove probabilistic results on limits of an infinite sequence of random variables.
• discuss the differences between the convergence concepts.

Sandsynlighedsteori (Sand) - alternatively Mål- og integralteori (MI) from previous years.

Academic qualifications equivalent to a BSc degree is recommended.
5 hours of lectures and 3 hours of exercises per week for 7 weeks.
The course is similar to Stochastic Processes 2 (NMAB15025U)
• Category
• Hours
• Lectures
• 35
• Preparation
• 146
• Theory exercises
• 21
• Exam
• 4
• Total
• 206
Written
Oral
Continuous feedback during the course of the semester

Written feedback in the form of comments to the compulsory
assignements.

Oral feedback during exercise classes, as a response to the
contribution of the students to the solution process of the
exercises.

Credit
7,5 ECTS
Type of assessment
Written examination, 4 hours under invigilation
Type of assessment details
Skriftlig prøve
Exam registration requirements

Approval of two assignments during the course is required to register for the exam.

Aid
All aids allowed
Marking scale