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Syllabus for

Academic year
MVE410 - Weak convergence
Svag konvergens
 
Syllabus adopted 2019-02-22 by Head of Programme (or corresponding)
Owner: MPENM
7,5 Credits
Grading: UG - Pass, Fail
Education cycle: Second-cycle
Major subject: Mathematics
Department: 11 - MATHEMATICAL SCIENCES

This course round is given every other year. Is given 2020/2021 but not 2021/2022


Teaching language: English
Application code: 20143
Open for exchange students: Yes

Module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0114 Examination 7,5c Grading: UG   7,5c    

In programs

MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 2 (elective)
MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 1 (elective)

Examiner:

Michael Björklund


Eligibility

General entry requirements for Master's level (second cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

English 6 (or by other approved means with the equivalent proficiency level)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

The prerequisite for the course is the equivalent of the courses MVE140 Foundations of probability theory and TMV100 Integration theory.

Aim

The aim of the course is to introduce the main concepts and give the proofs of the key results of the theory of weak convergence of probability measures on such metric spaces as the space of continuous trajectories and the space of cadlag trajectories.

Learning outcomes (after completion of the course the student should be able to)

After having taken the course, one should be able to
- explain the details of the proofs of the main theorems covered in the course, especially the Functional Central Limit Theorem,
- solve the exercises given in the compendium,
- demonstrate understanding of the key concepts and ideas concerning the weak convergence of probability measures, such as tightness and Skorokhod convergence.

Content

This course deals with weak convergence of probability measures on Polish spaces. Here the principal examples of Polish spaces are the space C = C[0, 1] of continuous trajectories and the space D = D[0, 1] of cadlag trajectories. Main topics:
- Portmanteau and mapping theorems
- Tightness and Prokhorov theorem
- Functional central limit theorems on C and D
- Empirical distribution functions and the Brownian bridge
- Weak convergence on D[0,∞)

Organisation

Lectures. Reading assignments.

Literature

Convergence of Probability Measures 2nd ed by Patrick Billingsley. A compendium downloadable from the course webpage.

Examination including compulsory elements

Oral and/or written examination.


Published: Mon 28 Nov 2016.