MATH 587 -- Advanced Probability Theory I

Fall 2013 -- Course syllabus


Louigi Addario-Berry
| | Tel: (514) 398-3831 (office) | 1219 Burnside Hall | Office Hours: Monday and Wednesday, 9:30-11:00 or by appointment

Time and Location  

TTh 2:35-3:55 | Burnside Hall 1205 |

Course book  

David Williams, Probability with Martingales.

Supplemental notes may be posted for some topics.

Notes from the first class.

Two proofs of Scheffé's lemma. (The first is the Lp formulation but is a bit terse; the second is just for L1 but has a little more detail.)

Notes on the weak and strong laws of large numbers.

Related writing  

Tao's review of probability theory and some of the links therein (notably to the Fubini-Tonelli theorem and the Radon-Nikodym derivative). Also, Tao's review of measure and integration theory.

The following notes on stochastic processes may also be of interest.


Will be posted here. Assignment 1 | Assignment 2 | Assignment 3 (see this for some of the exercises) | Solutions to Assignment 3 | Assignments 4 and 5.

Course Outline  

PDF version.

Lecture Schedule  

  1. Intro; Branching process example.
  2. What is a probability space? Algebras, sigma-algebras and measures.
  3. Dynkin's lemma; uniqueness of extension.
    [Supplementary reading: A non-measurable set from coin flips (optional).]
  4. Caratheodory's extension theorem.
  5. Lebesgue measure; completion of a measure space.
  6. Events; concrete probability spaces; Fatou for sets; Borel--Cantelli Lemma.
  7. Measurable functions; random varibales; generated sigma-algebras.
  8. Distribution functions; Lebesgue-Stieltjes measure; "Granularity".
  9. Indepenence: events, random variables, sigma-algebras; sufficient conditions for independence; Second Borel--Cantelli lemma.
  10. Sequences of independent random variables; Markov chains by hand; an aside on stochastic processes and continuity.
    [Supplementary reading: The Kolmogorov extension theorem (optional)]
  11. Integration: definitions, basic properties; monotone convergence theorem.
  12. Fatou's lemma; Dominated convergence; Riemann versus Lebesgue integration.
  13. Integrals over subsets, change of measure, and a first glimpse of Radon--Nikodym. PDFs and the "elementary formula" for expectation.
  14. Expectation: Markov, Jensen, Hölder, Minkowski, and L^p-spaces.
  15. Cauchy--Schwartz and the metric structure of L^2.
  16. Laws of large numbers 1.
  17. Laws of large numbers 2.
    [Supplementary notes to be posted.]
  18. Product measure; the monotone class theorem.
  19. Fubini's theorem.
  20. Independence and product measure; product spaces and separability; cylinder sets and an aside on continuity.
    [Supplementary reading: Cylinder sets and continuity (required, except for the section on Kolmogorov extension theorem).]
  21. Conditional expectation: the defining property, existence, almost sure uniqueness.
  22. Fundamental properties of conditional expectation.
  23. Conditional PDFs, conditional probabilities, conditional distributions.
    [Supplementary reading: Conditional distributions (required).]
  24. A second glimpse of Radon--Nikodym, and a first glimpse of Martingales.
  25. Review/spare.

Other resources  

Words of wisdom from Fields medalist Terence Tao, for graduate mathematics students:

George Polya's How to solve it and the Wikipedia version

Slightly off-base: Tim Gowers' mathematical discussions.

Additional Information  

In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.

McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see ) for more information).