On this page you will find brief reports from the lectures.
Monday, August 22nd: I first said a few words about the course and then continued with a quick repetition of Boolean set operations (union, intersection, and set-theoretic difference) with particular emphasis on distributive laws and the laws of De Morgan. Then I introduced \(\sigma\)-algebras and probability measures, proving Remark 1.2, Proposition 1.9, and Proposition 1.10 along the way. I postponed the material on algebras and monotone classes till next time (in particular the quite heavy proof of the Monotone Class Theorem 1.5) As I used the blackboard, there are no notes, but these notes from 2020 cover most of the same material.
Wednesday, August 24th: In this lecture, I covered the rest of sections 1.1-1.2. I first introduced \(\sigma\)-algebras generated by families of sets, mentioning Borels and cylinder sets as typical examples. I then introduced algebras and monotone classes and proved that a monotone class that is also an algebra, is in fact a \(\sigma\)-algebra. This was then used to prove the Monotone Class Theorem with a proof that is untypically technical for this course. Next time, I continue with sections 1.4 and 1.5, and perhaps even a little bit of 2.1. Although I skip (what I haven't already covered of) Section 1.3, I encourage you to study the examples there on your own. Here are some notes from 2020 that cover much of the same material.
Monday, August 29th: Before the break, I covered sections 1.4 and 1.5, following the book quite closely. After the break, I started Section 2.1, expanding a little on the quite terse treatment in the book. I stated (and partly proved) that a function \(X:\Omega\to\mathbb{R}\)is a random variable if and only if one of the following three conditions is satisfied for all \(\alpha\)
(i) \(\{\omega:X(\omega)<\alpha\}\)
(ii) \(\{\omega:X(\omega)\geq\alpha\}\)
(iii) \(\{\omega:X(\omega)>\alpha\}\)
I also showed that if \(X\)is a random variable, then \(X^{-1}(B)\in\mathcal{A}\)for all Borel sets \(B\)(Proposition 2.8). I also proved part (ii) of Exercise 2.1 and will prove (iii) next time. Next time, I shall also introduce distribution functions. Unfortunately, I don't have any notes in English for Section 2.1t, but here are some notes in Norwegian.
Wednesday, August 31st: Before the break, I proved part c) of Exercise 2.2, introduced distribution functions and proved Proposition 2.5.
After the break, I did problems. I spent most of the time on Exercise 1.2, not actually solving it, but working out a strategy for proving that \(\sigma(\mathcal{G})=\sigma(\mathcal{F})\). I then used this strategy to prove that the Borel \(\sigma\)-algebra equals the \(\sigma\)-algebra generated by the open interval (this is the argument sketched at the bottom of page 3) and also to solve part a) of the problem. I also solved problems 1.6, 1.16, 1.22, and 1.23. Notes.
Monday, September 5th: I first finished the introduction to distribution functions by proving a variant of Proposition 2.6. I then introduced distributions and proved Proposition 2.10 (Proposition 2.8 we have covered earlier). I skipped section 2.2 on existence of random variables and continued to Section 2.3 on independence. Most of the time here was spent on the quite long and intricate proof of Theorem 2.20. Next time there is no physical lecture, but I'll make a video covering as much as possible of Section 2.5 (and another video covering the problems).
Wednesday, September 7th: Prerecorded lectures:
Section 2.5: Introduction to expectations of discrete random variables. Video. Notes
Problems from the textbook: Video. Notes
Problem on alternative descriptions of random variables. Video. Notes
Problem on \(1/X(\omega)\). Video. Notes.
Problem on inverse images: Video. Notes
Monday, September 12th: Lectured on Section 3.1 on expectations of general random variables, but didn't get very far. Next time we shall start with Theorem 3.5 on the expectation of independent random variables. I also hope to cover the main ideas in Section 3.2. Here are notes from a previous year that covers most of the material in this lecture.
Wednesday, September 14th: Before the break, I finished Section 3.1 by proving Theorem 3.5 and going quickly through Section 3.2. After the break I did problems 2.10, 2.16, 2.20, 2.30, 2.37. Next time, we shall prove some useful inequalities in sections 3.3 and 3.4. Perhaps we shall even have time to begin Chapter 4. Here are some notes from a previous year (but I did mainly different problems then!)
Monday, September 19th: Proved the inequalities of Chebyshev, Schwarz, Jensen, and Lyapounov in sections 3.3 and 3.4. Will start Chapter 4 on Wednesday. Here are notes from an earlier year covering most of the material.
Wednesday, September 21st: Before the break, I covered Section 4.1 up to and including Example 4.5.1. After the break, I first apologized for giving too many hard problems for one week, and the solved problems 2.40, 2.41, 3.3, and 3.4. Here is note on Abel summation (which is useful for problems 3.4 and 3.6), and here are some notes from earlier years: Problems 2.41, 2.42. Problems 3.3, 3.4 (sketchy), 3.5. Problems 2.40, 2.41, 2.43 (in Norwegian). Problems 3.3, 3.4, 3.5, 3.6 (in Norwegian)
Monday, September 26th: Introduced limsup and liminf for sets, proved the Borel-Cantelli Lemma and Theorem 4.10. Spent a little extra time on sorting out the connection between infinite sums and infinite products that is needed in the proof of second part of the Borel-Cantelli Lemma. These notes from an earlier year covers the same material plus a little bit more.
Wednesday, September 28th: Before the break, I proved the Monotone Convergence Theorem and Fatou's Lemma (here are some notes from an earlier year). After the break, I concentrated on problems 3.13, 3.19 and the first problem from the assignment from 2019 (see the notes above and this solution of the assignment).
Monday, October 3rd: As I had a fever, the physical class was cancelled and replaced by the three videos below:
Dominated Convergence Theorem (section 4.2): Notes. Video.
Laws of Large Numbers (section 5.1): Notes. Video.
\(\sigma\)-algebras as information (section 5.4): Notes. Video.
Wednesday, October 5th: The physical class is cancelled due to illness and replaced by the videos below:
Independence of \(\sigma\)-algebras (section 5.5): Notes. Video.
Zero-One Laws (section 5.6, new version): Notes. Video.
Problems from the textbook: 4.7, 4.9, 4.11, 4.13, 4.23. Notes. Video.
Remaning problems: Notes. Video.
Monday, October 10th: Talked a little about expectations of complex random variables before I defined characteristics functions. I then covered Theorem 6.3, Theorem 6.4(i), and Theorem 6.5. I also computed the characteristic function of a gaussian rndom variable (Example 6.8.1). Next time we shall look at subsection 6.1.1, and it may be a good idea to take a look at this note on Fourier inversion first. Here are notes from an earlier year covering (more than) today's material: Notes 1. Notes 2.
Wednesday, October 12th: Talked informally about Fourier inversion and used it to prove Lévy's Inversion Theorem 6.11. After the break, I solved problems 5.11 and 5.12. Here are some notes from earlier years: Lévy's Inversion Theorem, Problem 5.10, Problem 5.11, Problem 5.11 and 5.12 (in Norwegian).
Monday, October 17th: Proved Theorem 6.17 and Proposition 6.15 (which we shall use next time). Stated Helly's Theorem (6.19), but didn't have time to prove it. Next time, I'll prove Helly's Theorem, sketch the proof of Theorem 6.22, and then continue to section 6.3, skipping the rest of section 6.2.
Wednesday, October 19th: Before the break, I proved Helly's Theorem, introduced the important notion of tightness, and proved Theorem 6.22. Next time, I shall continue with Section 6.3. After the break, I solved Problems 5.8, 6.2, 6.3, 6.5, and quickly sketched the solution to the problem from the trial exam. Here are some notes from earlier years: Problems 6.2, 6.3, 6.5. Trial Exam.
Monday, October 24th: Today I covered Proposition 6.29, Lemma 6.30, Theorem 6.32 (Lévy's Continuity Theorem), Lemma 6.34, and Theorem 6.37 (simple form of the Central Limit Theorem). Next time I shall continue with Theorem 6.38 (Lyapounov's version of the Central Limit Theorem). Notes from an earlier year: Proposition 6.29 and Lemma 6.30. Lévy's Continuity Theorem and Lemma 6.34. Simple form of CLT.
Wednesday, October 26th: Before the break, I proved Lyapounov's version of the Central Limit Theorem (Theorem 6.38). The proof in the book is rather tersely written, and I tried to organize the proof in a way that made it possible to cover most of the details without losing touch of the central ideas (here are some notes from 2020 following the same outline). After the break, I did problems 6.6 and 6.9. Here are some notes from earlier years: Problems 6.6 and 6.8 (in Norwegian). Problems 6.9, 6.21, 6.22.
Next time, I'll stop by Section 7.1 and 7.4 to pick some basic properties of stochastic processes, filtrations, and stopping times before we proceed to Chapter 8. The intuition behind filtrations and stopping times can be hard to grasp, and it may be helpful to take a look at these notes (even though they will not be covered in class).
Monday, October 31st: Today I covered Section 7.4 on stopping times and then continued with Chapter 8 where I just managed to finish Proposition 8.2. Next time, I'll continue with the rest of 8.1 and 8.2 (section 8.3 is not on the syllabus). If you find stopping times mysterious (and they are!), I still recommend my note on the subject.
Wednesday, November 2nd: Before the break I continued the study of conditional expectations by going through Example 8.2.1 and proving all parts of Theorem 8.3 except (viii), which I shall do next time. After the break, I did Exam 2019, problem 2, and Exam 2020, problem 1.
Monday, November 7th: Finished section 8.2 on conditional expectations by proving Theorem 8.3(viii), Theorem 8.4 and Theorem 8.7 (Jensen's inequality for conditional expectations). With Jensen's inequality I chose a different proof than the book. After the break, I started Chapter 9 by introducing martingales, submartingales, and supermartingales and proving (some of) Proposition 9.4 and 9.6. Here and here are notes from an earlier year covering part of the theory.
Wednesday, November 9th: Before the break I did a variant of Exercise 1 (introducing the notation \(\Delta X_n=X_{n+1}-X_n\) which I find quite helpful) and then theorems 9.7 (Doob Decomposition) and 9.8. After the break I did problem 4 from the exam 2019, plus problem 7.14 and 7.15. Here are notes from an earlier year covering most of the theory. Solutions to the exam from 2019 are here.
Monday, November 14th: No physical lecture, only prerecorded videos. The plan is to make five relatively short ones:
Gambling and martingales: Notes. Video
Martingale inequalities: Notes. Video
Upcrossing inequality: Notes. Video
Martingale convergence: Notes. Video
Wednesday, November 16th: Before the break I introduced uniform integrability and proved Remark 9.27 and Exercise(!) 9.14 (this result is so central in the sequel that it feels unfair to leave it as an exercise). After the break I solved exercise 9.5 in a very clumsy way. Notes from earlier years: Uniform integrability (in Norwegian). Problems from Chapter 8. Problem 9.3 and 9.5.
Monday, November 21st: I proved Theorems 9.28, 9.29, 9.30, and 9.31. Next time I shall finish the syllabus by talking about the first application in Section 9.5 (the existence of conditional expectations). Notes of some of the material (in Norwegian).
Wednesday, November 23rd: Finished the syllabus by going through the first application in 9.5, then did all this weeks problems. You'll finds the solutions here: Trial Exam 1. Trial Exam 2. Exam 2019. Exam 2020. Remember that the reviews next week are both in room 1119 (but at the usual time).
Monday, November 28th: Reviewed \(\sigma\)-algebras, algebras, monotone classes, filtrations, probability measures, random variables, expectations, conditional expectations, modes of convergence, and convergence theorems (Monotone Convergence Theorem, Fatou's Lemma, Dominated Convergence Theorem, Uniform integrability). Here are reviews from an earlier year (the order is slightly different): Notes 1, Notes 2.