Kursöversikt
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 | 4.2 | 11 | Convergence of martingales (a.s.) |
| 10 | 22/4 | 4.4 | 14.6, 14.11 | Convergence of martingales (L^p) |
| 11 | 23/4 | 4.6 | 13, 14 | Convergence of martingales (L^1), Uniform integrability |
| 12 | 28/4 | 4.6 | 14 | UI martingales |
| 13 | 29/4 | 4.7 | 14.4, 14.12 | Reverse & product martingales |
| 14 | 5/5 | 4.5 | 12.1, 12.6-7, 12.11-15 | Levy's Borel-Cantelli lemma |
| 15 | 6/5 | Exchangeable processes and de Finettis theorem | ||
| 16 | 7/5 | Exchangeable processes and de Finettis theorem | ||
| 17 | 12/5 | Stationary processes and ergodic theorems | ||
| 18 | 13/5 | Stationary processes and ergodic theorems | ||
| 19 | 19/5 | Brownian motion: basic properties | ||
| 20 | 20/5 | Brownian motion: further properties | ||
| 21 | 21/5 | Repetition / extra time | ||
| 30/5 | EXAM |
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.
Homework
There will be two homework sheets to hand in for grading. These will give up to 4 bonus marks for the exam. 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
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).
Old exams
Exam August 2022 and solutions.
Exam August 2024 and solutions
Kurssammanfattning:
| Datum | Information | Sista inlämningsdatum |
|---|---|---|