MSF500/MVE410, Weak Convergence, Spring 19


Welcome to the course! The schedule can be found here.

We will meet [in class] 7 times (2 hours each time) over the coming weeks. However, each week, I will have office hours during which I am open to discuss the problems in the home assignments as well as the theory needed to solve them.

My office is H4017, and the agreed upon meeting times are:

Tuesdays, 15.15-17.00 [not on May 7, as I'm giving the analysis seminar in MVL14 then]

[Thursday meetings are cancelled due to low attendance].

You are always welcome to send me questions, or come by, but please send an email first so that I can make sure that I'm available.


Docent Michael Björklund

Course literature

Lecture notes can be found here.

Alternative sources:

Probability-inclined people tend to like the book "Convergence of Probability Measures" by Billingsley. Previous years courses have been based on this book. Lectures notes from these years can be found here.

Analysis-inclined people do not have a book on this topic. A (good) attempt to please this crowd was made by Stroock with his book "Probability theory - an analytic view". Some parts of my lecture notes are based on this (quite readable) book.

Courses on "Weak convergence of probability measures on metric spaces" are continually given all over the world. Many lecturers produce their own versions of the standard material. Feel free to use Google to find your favorite take. 


Lecture I: Hellinger distances and decorrelation.

Abstract: We introduce the Hellinger distance between bounded and positive measures, and show that it is
lower semicontinuous wrt narrow (weak*) convergence. As an application, we give a criterion for decorrelation
of two sequences in the Wiener space in terms of singularity of their associated spectral measures. The classical
Wiener-Wintner Theorem will be a corollary.

Tools to be developed: Narrow convergence, spectral measures (Herglotz's Theorem), Wiener's Lemma.


Lecture II: The role of tightness.

We begin to prepare for the proof of Donsker's Theorem and the construction of the Wiener measure by going through some rather dry material pertaining to narrow convergence and tightness in separable metric spaces. If time permits, some Arzela-Ascoli theory will be discussed.

Tools to be developed: Portmanteau's Lemma, tightness.


Lecture III: The Wiener measure

Abstract: We specialize our theory so far to path spaces (=continuous maps from [0,\infty) into a Banach space) and prove various criteria for tightness (a.k.a. pre-compactness) for a given subset of Borel probability measures. Once this is done, we shall prove Donsker's Theorem, providing a construction of the Wiener measure.

Tools to be developed: Besov spaces, Kolmogorov's tightness criteria.


Lecture IV: Levy processes - Infinite divisibility and Levy-Khinchin's Theorem

Abstract: We discuss infinitely divisible probability measures on the real line and prove Levy-Khinchin's representation theorem for such measures. This is the first step towards constructing general Levy measures on spaces of Cadlag functions.


Lecture V/VI: Crash course in ergodic theory

Abstract: We outline some basic results in ergodic theory where weak convergence/compactness is used -
existence of invariant measures, ergodic measures, unique ergodicity.

Course requirements

The learning goals of the course can be found in the course plan.


Can be found in the lecture notes.


The course is examined through home assignments and an oral exam.

A total of 100 points can be awarded on the home assignments (which are to be handed in individually, but you are free to work together). A minimum of 60 points is needed to qualify for the oral exam.

The final grade will depend on your home assignments and your performance during the oral exam.

Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.

Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal, where you also can read about what rules apply to examination at University of Gothenburg.

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.

At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination:
Exams are reviewed and retrieved at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.