
MATH 587  Advanced Probability Theory I
Fall 2023  Course syllabus



Course Outline

PDF version.

Professor

Louigi AddarioBerry 
louigi.addario@mcgill.ca 
Tel: (514) 3983831 (office) 
1223 Burnside Hall 
Office Hours: Wednesday, 14:3515:55 or by appointment

Time and Location

TTh 2:353:55 
MH Wong 1050 

Course book

My course notes (linked below).

Supplemental materials

Written material (not by me):
Rick Durrett, Probability, Theory and Examples, 5th edition.
Patrick Billingsley, Probability and Measure. (Ridiculously expensive.)
James Norris, Lecture notes on probability.
Franco Vivaldi, Mathematical writing for undergraduate students. N.B. AND GRADUATE STUDENTS.
Videos:
Claudio Landim, Lectures on measure theory
Hao Wu, Martingales and Markov Processes.
Fematika, Measure theory lectures made by a highschool student from Ohio named Lucas.
Written material (by me):
 My course notes.
 My handwritten lecture notes (large files).
 Random walks and branching processes; a story to motivate the development of the theory.

Assignments

Will be posted on MyCourses.
Solution template (LaTeX).

Lecture Schedule (tentative)

 Lecture 1. Introduction; start of measure theory development. Reference: Outline, my course notes.
 Lecture 2. Carathéodory extension theorem. Reference: my course notes.
 Lecture 3. Dynkin's lemma and Dynkin's theorem; uniqueness of extension. Reference: my course notes.
 Lecture 4. Stieltjes functions and cumulative distribution functions. Existence of measures (in particular Lebesgue measure). Reference: my course notes.
 Lecture 5. Independent events, independent sigmafields; BorelCantelli lemmas. Reference: my notes.
 Lecture 6. Random variables and measurable maps; countable operations with random variables give random variables. Generated sigmafields, independence of random variables, distribution of a random variable. Reference: my notes.
 Lecture 7. Existence of independent random variables with given distributions. Kolmogorov 01 law and examples. Reference: my notes.
 Lecture 8. Types of convergence: almost sure convergence, convergence in probability, convergence in distribution. Implications between types of convergence. Couplings; Skorohod representation theorem for real random variables. Reference: my notes.
 Lecture 9. Defining the integral: simple functions; nonnegative functions; L1 functions; linearity of expectation; monotone convergence theorem. Reference: my notes.
 Lecture 10. Inevitability of the definition of the integral. Almost everywhere equivalence. Fatou's lemma; dominated, bounded convergence theorem. Expectations and independence; factorization of expectations for products of independent random variables. Reference: my notes.
 Lecture 11. Monotone class theorem. Examples: the probabilistic method. Densities and the change of variables formula. Reference: my notes.
 Lecture 12. More change of variables examples; computations with random variables. Start of product spaces. Reference: my notes.
 Lecture 13. Product measures and Fubini's theorem; measurability of sections and marginals. Reference: my notes. Reference: my notes.
 Lecture 14. Sums of independent random variables; convolution of CDFs; Markov's inequality and its variants; the L2 weak law of large numbers. Reference: my notes.
 Lecture 15. Chernoff bound and concentration of measure. Weak law of large numbers for L1 random variables. Start of Lacunary strong law of large numbers. Reference: my notes.
 Lecture 16. Lacunary strong law of large numbers and extension to strong law of large numbers. Reference: my notes.
 Lecture 17. Convexity: Jensen's inequality, Hölder and Minkowski's inequalities. Monotonicity of Lp norms; relations between convergence in Lp and almost sure convergence/convergence in probability. Reference: my notes.


