P R O B A B I L I T Y,    F a l l    2 0 2 3

Lectures: Monday, Wednesday, 11:00-12:15pm, in Warren Weaver Hall 201.

Lecturer: Paul Bourgade, office hours Thursday 11.00am-12.00, you also can email me (bourgade@cims.nyu.edu) to set up an appointment or just drop by (WWH 629).

Course description: First semester in an annual sequence of Probability Theory, aimed primarily for Ph.D. students. Topics include laws of large numbers, weak convergence, central limit theorems, conditional expectation, martingales and Markov chains.

Prerequisites: A first course in probability, familiarity with Lebesgue integral, strong analysis level.

Textbooks: Our reference text will be Probability Theory, by S.R.S. Varadhan.

Homework: Every Wednesday for the next Wednesday.

Grading: problem sets (50%) and final (50%).

A tentative schedule for this course is:

• Sep. 6. Measure theory: Introduction, Caratheodory extension theorem (existence).
• Sep. 11. Measure theory: Caratheodory extension theorem (uniqueness), Lebesgue's characterization for ℝ, Integration (random variables).
• Sep. 13. Measure theory: Integration (convergence theorems).
• Sep. 18. Measure theory: Transformations, product spaces.
• Sep. 20. Measure theory: Distributions and expectations.
• Sep. 25. Weak convergence: Characteristic functions, Lévy's theorem
• Sep. 27. Weak convergence: Bochner's theorem.
• Oct. 2. Independent sums: Convolutions, weak law of large numbers.
• Oct 4. Independent sums: Central limit theorem.
• Oct 10. Independent sums: Borel-Cantelli, 0-1 laws.
• Oct 11. Independent sums: Weak and strong law of large numbers.
• Oct 16. Independent sums: Accompanying laws and infinite divisibility.
• Oct. 18. Dependent random variables: Conditioning.
• Oct 23. Dependent random variables: The Radon-Nikodym Theorem.
• Oct. 25. Dependent random variables: Conditional expectation and conditional probability.
• Oct 30. Dependent random variables: Markov chains 1.
• Nov 1. Dependent random variables: Markov chains 2.
• Nov 6. Dependent random variables: Markov chains 3.
• Nov. 8. Dependent random variables: Markov chains 4.
• Nov 13. Dependent random variables: Markov chains 5.
• Nov. 15. Martingales 1.
• Nov 20. Martingales 2.
• Nov. 27. Martingales 3.
• Nov. 29. Martingales 4.
• Dec 4. Martingales 5.
• Dec. 6. Stationary processes 1: ergodic theorems.
• Dec 11 Stationary processes 2: stationary measures.
• Dec. 13. Stationary processes 3: the Markov case.
• Dec. 19, Tuesday, 12-2pm, room 201. Final exam.

• Problem set 1. Tex file.
• Problem set 2. Tex file.
• Problem set 3. Tex file.
• Problem set 4. Tex file.
• Problem set 5. Tex file.
• Problem set 6. Tex file.
• Problem set 7. Tex file.
• Problem set 8. Tex file.
• Problem set 9. Tex file.
• Problem set 10. Tex file.
• Problem set 11. Tex file.
• Problem set 12. Tex file.