# Advanced Probability Theory 1 (VidSand1)

### 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

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

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.

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

Academic qualifications equivalent to a BSc degree is recommended.

The course is similar to Stochastic Processes 2 (NMAB15025U)

Written
Oral
Continuous feedback during the course

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
Type of assessment details
Skriftlig prøve
Aid
All aids allowed
Marking scale
Censorship form
External censorship
##### 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
• 146
• Theory exercises
• 21
• Exam
• 4
• English
• 206

### Kursusinformation

Language
English
Course number
NMAK11003U
ECTS
7,5 ECTS
Programme level
Full Degree Master
Duration

1 block

Placement
Block 1
Schedulegroup
B
Capacity
No limit
The number of seats may be reduced in the late registration period
Studyboard
Study Board of Mathematics and Computer Science
##### Contracting department
• Department of Mathematical Sciences
##### Contracting faculty
• Faculty of Science
##### Course Coordinator
• Ernst Hansen   (8-69766c6572776972447165786c326f7932686f)
phone 35 32 07 73, office 04.3.12,
Saved on the 28-02-2022

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Courseinformation of students