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 (meaning usually kl 15.15 Wednesdays and Fridays). Recordings of lectures from 2020 are available below. This year's material may differ somewhat from the recordings, but not in a major way. The lectures will be more focused on examples than the recordings, and thus treat the "theory" parts a bit more briefly.

The course will mostly follow the book Probability and random processes by Grimmett and Stirzaker (G&S), selected parts of chapters 7 -- 12. 

For the last lectures, we will follow Chapter 1 of the book Brownian Motion by Mörters and Peres. The full book is available at the homepage of Peter Mörters, and also here.

Lectures

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

Lecture	Day	Sections	Contents and link to video/notes
1	23/3	G&S: 7.3, 7.9	Repetition, inequalities, conditional expectation, Borel--Cantelli lemmas
2	25/3	G&S: 7.2, 7.10	Modes of convergence and uniform integrability
	30/3		no lecture
3	1/4	G&S: 8.2; 9.1-3, 6	Weakly and strongly stationary processes
4	6/4	G&S: 9.5	Ergodic theorem for stationary processes
5	8/4	notes	De Finetti's theorem for exchangeable processes
6	20/4	G&S: 10.1-2	Renewal processes
7	22/4	G&S: 10.3-5	Renewal processes continued
8	27/4		not queuing theory
9	29/4	G&S: 12.1 + 7.8	Martingales, convergence in L^2
10	4/5	G&S: 12.2, 12.1.10, 12.8.1	Doob's decomposition and Hoeffding's inequality
11	6/5	G&S: 12.3	Convergence in L^1 and Doob's martingale
12	11/5	G&S: 12.4-5	Optional sampling theorem
13	13/5	G&S: 12.6-7	Maximal inequalities and backward martingales
14	18/5	M&P: 1.1.(1, 3), 1.3	Brownian motion: basic properties
15	20/5	M&P: 1.1.2, 1.3	Brownian motion: further properties
	25/5		Possible extra time / repetition

 

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Recommended exercises

There are worked solutions to the exercises in G&S in the book One Thousand Exercises in Probability by the same authors.

Lectures	Exercises/Problems
1-2	G&S Ch. 7: 1.5, 2.1, 2.2, 2.7, 3.1, 3.2, 3.3, 3.4, 3.9, 4.1, 5.1,  9.1, 10.1, 10.6, 11.34
3-4	G&S Ch. 9: 1.2, 1.4, 2.1, 2.2, 3.2, 3.3, 5.1, 5.2, 6.2, 6.4, 7.9, 7.11, 7.12, 7.13
5	on the separate sheet
6-8	G&S Ch. 10: 1.1, 1.4, 2.1, 2.2, 2.3, 3.4, 5.3, 6.1, 6.16ab
9-13	G&S Ch. 7: 7.1, 8.3, 11.27 G&S Ch. 12: 1.5, 1.7, 1.8, 2.1, 3.4, 4.1, 4.5, 5.4, 7.3, 9.6, 9.7, 9.11, 9.13, 9.16
14-15	M&P: 1.1, 1.3, 1.5, 1.6, 1.12

 

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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 (each sheet is marked out of 50, ie 5 points per question, then the total score from the 2 sheets is mapped to bonus points using the limits 20, 40, 60, 80). The sheets will be due for hand-in on May 4th and May 25th respectively.

Here is the first homework and solutions

Here is the second homework and 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 the theorems which may occur on the final exam.

Here is the June 2022 exam and the solutions

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Old/mock exams

Last time the course was held, in 2020, the exam was oral due to the pandemic. You can see the material used for the oral exam here.

Here is a mock exam for the course, of roughly half the size as the actual exam will be. It concerns only material before the chapter on martingales. Solutions are here. 

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