
MATH 587  Advanced Probability Theory I
Fall 2013  Course syllabus



Professor

Louigi AddarioBerry 
louigi@math.mcgill.ca 
Tel: (514) 3983831 (office) 
1219 Burnside Hall 
Office Hours: Monday and Wednesday, 9:3011:00 or by appointment

Time and Location

TTh 2:353:55 
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 FubiniTonelli theorem and the RadonNikodym derivative). Also, Tao's review of measure and integration theory.
The following notes on stochastic processes may
also be of interest.

Assignments

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

 Intro; Branching process example.
 What is a probability space? Algebras, sigmaalgebras and measures.
 Dynkin's lemma; uniqueness of extension.
[Supplementary reading: A nonmeasurable set from coin flips (optional).]
 Caratheodory's extension theorem.
 Lebesgue measure; completion of a measure space.
 Events; concrete probability spaces; Fatou for sets; BorelCantelli Lemma.
 Measurable functions; random varibales; generated sigmaalgebras.
 Distribution functions; LebesgueStieltjes measure; "Granularity".
 Indepenence: events, random variables, sigmaalgebras; sufficient conditions for independence; Second BorelCantelli lemma.
 Sequences of independent random variables; Markov chains by hand; an aside on stochastic processes and continuity.
[Supplementary reading: The Kolmogorov extension theorem (optional)]
 Integration: definitions, basic properties; monotone convergence theorem.
 Fatou's lemma; Dominated convergence; Riemann versus Lebesgue integration.
 Integrals over subsets, change of measure, and a first glimpse of RadonNikodym. PDFs and the "elementary formula" for expectation.
 Expectation: Markov, Jensen, Hölder, Minkowski, and L^pspaces.
 CauchySchwartz and the metric structure of L^2.
 Laws of large numbers 1.
 Laws of large numbers 2.
[Supplementary notes to be posted.]
 Product measure; the monotone class theorem.
 Fubini's theorem.
 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).]
 Conditional expectation: the defining property, existence, almost sure uniqueness.
 Fundamental properties of conditional expectation.
 Conditional PDFs, conditional probabilities, conditional distributions.
[Supplementary reading:
Conditional distributions (required).]
 A second glimpse of RadonNikodym, and a first glimpse of Martingales.
 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 offbase: 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 www.mcgill.ca/students/srr/honest/ ) for more information).


