This page contains the program of the course: lectures, exercises and homework. Other information, such as learning outcomes, are in a separate course PM.

Here is the May 31st exam and solutions.

Program

The course will be given in person according to the schedule in TimeEdit 

Literature

The course will mostly follow the book Probability with Martingales by David Williams. Another very useful source is Probability and random processes by Grimmett and Stirzaker (G&S), chapters 7 -- 12. We will also use small parts of the book Brownian Motion by Mörters and Peres. The full book is available at the homepage of Peter Mörters, and also here. Finally, parts of the course will use Rick Durrett's Probability: theory and examples which is available at Durrett's homepage.

The only book I know that fully covers all parts of the course is Foundations of Modern Probability by Olav Kallenberg, but it is more a reference work rather than a book to learn from.

Lectures

The following schedule should be regarded as preliminary, and may come to be updated as the course progresses.

Lecture	Day	Sections	Contents
1	19/3	DW ch 1-3	Fundamentals of measure-theoretic probability
2	21/3	DW ch 4	Borel--Cantelli lemmas, Kolmogorov zero-one law
3	26/3	DW ch 6, 9	Modes of convergence; conditional expectation
4	23/3	DW ch 9, 10	Martingales, convergence in L^2
5	9/4	DW ch 10, 11	Previsible processes, Convergence of martingales
6	11/4	DW ch 11, 12	Convergence of martingales in L^2
7	18/4	DW ch 12, 10	L^2 mg ctd; optional stopping
8	23/4	DW ch 10, 13	OST; uniform integrability
9	25/4	DW ch 14	UI martingales, reverse martingales
10	2/5	Durrett 4.7	Reverse martingales and exchangeability
11	7/5	Durrett 4.7	Exchangeable processes and de Finettis theorem
12	8/5	GS 9.5	Stationary processes and ergodic theorems
13	14/5	GS 9.5	Stationary processes and ergodic theorems
14	16/5	GS 9.5	Stationary processes and ergodic theorems
15	21/5	MP ch 1	Brownian motion: basic properties
16	23/5	MP ch 1-2	Brownian motion: further properties
	31/5		EXAM

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Exercises

Here is a sheet of exercises, sourced from various places. The list will grow as the course progresses.

Here are solutions to (most of) the exercises.

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Homework

There will be two homework sheets to hand in for grading. These will give up to 4 bonus marks for the exam (the first sheet is marked out of 50, the second out of 35, then the total score from the 2 sheets is mapped to bonus points using the limits 17, 34, 51, 68).

The hand-ins will be due on the 2nd May and the 21st May.

Here is the first homework, due 2nd May. Here are the solutions.

Here is the second homework, due 28th May. Solutions.

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Examination

There will be a final exam in June. It will consist of between 6 and 8 questions, and will be marked out of 50. The only allowed tool for the exam is a pen! (No calculator, no notes, etc.) For Chalmers, the cut-offs are 20, 30 and 40 points including any bonus points (for grades 3, 4, 5) while for GU the cut-offs are 20 and 35 including any bonus points (for grades G and VG).

On the exam will be some "known" problems (from the books or lectures), some "unknown" problems, and some "bookwork" (giving definitions, theorems, and proofs).

Here is a list of proofs which may be asked on the exam.

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Old/mock exams

Exam June 2022 and solutions.

Exam August 2022 and solutions.

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Old videos and notes

There are some videos and notes from the pandemic years on this page. They overlap with the current course but only to a certain extent.

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