# Stochastic Processes 2

### Course 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.
Education

BSc Programme in Actuarial Mathematics

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.

5 hours of lectures and 3 hours of exercises per week for 7 weeks.

Mål- og integralteori (MI)

The course is equivalent to the course Advanced Probability Theory 1 (VidSand1) (NMAK11003U)

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.

ECTS
7,5 ECTS
Type of assessment
Written examination, 4 hours under invigilation
The course has been selected for ITX exam on Peter Bangs Vej.
Aid
All aids allowed

The University will make computers and power available to students taking written exams with invigilation in the University’s building on Peter Bangs Vej 36 (ITX). Students are therefore not permitted to bring their own computers, tablets or mobile phones. If textbooks and/or notes are permitted, according to the course description, these must be in paper format or on a USB flash drive.

Marking scale
Censorship form
No external censorship
One internal examiner.
##### Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Single subject courses (day)

• Category
• Hours
• Lectures
• 35
• Preparation
• 147
• Theory exercises
• 21
• Exam
• 3
• English
• 206