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

Program

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

Literature

The course will mostly follow the book Rick Durrett's Probability: theory and examples which is available at Durrett's homepage.

Other useful sources include Probability with Martingales by David Williams, and Probability and random processes by Grimmett and Stirzaker.

Lectures

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

Lecture	Day	Approx sections in Durrett	Approx sections in Williams	Contents
1	24/3	1.1 - 1.3	Ch 1 - 3	Fundamentals of measure-theoretic probability
2	25/3	2.1	Ch 3 - 4	Independence; Borel--Cantelli lemmas
3	26/3	2.3, 1.4 - 1.6	Ch 4 - 6	Borel--Cantelli lemmas ctd; expectation
4	31/3	here & there	Ch 6 for parts	Modes of convergence
5	1/4	here & there	Ch 6 for parts	Modes of convergence ctd
6	14/4	4.1	6.10 - 6.11, Ch 9	Conditional expectation, martingales
7	15/4	4.2, 4.8	10.3-4, 10.8-10.10	Martingales, optional stopping
8	16/4	4.8, 4.2	10.10, 10.12, 11	Optional stopping ctd; convergence of mg
9	21/4	4.4	14.6, 14.11	Convergence of martingales (a.s. and L^p)
10	22/4	4.6	13, 14	Convergence of martingales (L^1), Uniform integrability
11	23/4	4.6	14	UI martingales, Kolmogorov's 0-1 law
12	28/4	4.7	14.4, 14.12	Reverse & product martingales
13	29/4	4.5	12.1, 12.6-7, 12.11-15	Levy's Borel-Cantelli lemma
14	5/5			Exchangeable processes and de Finettis theorem
15	6/5			Exchangeable processes and de Finettis theorem
16	7/5			Stationary processes and ergodic theorems
17	12/5			Stationary processes and ergodic theorems
18	13/5			Brownian motion: basic properties
19	19/5			Brownian motion: further properties
20	20/5			Repetition / extra time
21	21/5			Repetition / extra time
	30/5			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 based on the total score on the two hand-ins. The maximum score is 70, the limits are: 1 pt for 15-28, 2 pts for 29-42, 3 pts for 43-56, 4 pts for 57-70.

To pass each homework you must also give a 15 minute oral presentation of your solution to one of the problems (I choose which one). 

Here is the first homework sheet, due april 29th at 15.15

The oral presentations for HW1 are planned for the 4th May - here is a link for booking: Book time with Jakob Björnberg: 15 minutes meeting If the 4th May does not work, email me and we can find another time.

Here is the second homework sheet, due May 20th at 15.15.

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Examination

There will be a final exam on May 30th. 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).

(Non)-proofs: 

Any result proved in the lectures can come as a question in the exam. Some results were stated in the lectures but not proved, and these will not be asked on the exam. These are (or include...):

• The pi-systems lemma (DW 1.6)
• The existence of expectations for non-negative and integrable random variables
• The monotone convergence theorem
• The change-of-variables formula for expectations (Du Thm 1.6.9)
• Dual characterization of weak convergence (Du Thm 3.2.9)
• Completeness of L^2 (DW 6.10)
• Hölder's inequality
• Kronecker's lemma

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Old exams

Exam June 2022 and solutions.

Exam August 2022 and solutions.

Exam May 2024 and solutions

Exam August 2024 and solutions

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