# 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 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
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
Censorship form
External censorship
Re-exam

Same as ordinary exam.

If the compulsory assignments were not approved before the ordinary exam they must be (re)submitted. They must be approved at the latest three weeks before the beginning of the re-exam week.

##### Criteria for exam assessment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.

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   (8-68756b6471766871437064776b316e7831676e)
phone 35 32 07 73, office 04.3.12,
Saved on the 28-02-2023

### Are you BA- or KA-student?

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Courseinformation of students