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

Lectures: Tuesday, 11:00am-12:50pm, in Warren Weaver Hall 1302.

Lecturer: Paul Bourgade, office hours Thursday 10.00am-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 Tuesday for the next Tuesday.

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

A tentative schedule for this course is:

• Sep. 6. Measure theory: Construction of measures.
• Sep. 13. Measure theory: Integration.
• Sep. 20. Measure theory: Transformations, product spaces, distributions and expectations.
• Sep. 27. Weak convergence.
• Oct 4. Independent sums: Convolutions, laws of large numbers.
• Oct 11. No class, Monday schedule.
• Oct. 18. Independent sums: Central limit theorem.
• Oct. 25. Dependent random variables: Conditioning, the Radon-Nikodym Theorem.
• Nov 1. Dependent random variables: Conditional expectation.
• Nov. 8. Dependent random variables: Markov chains 1.
• Nov. 15. Dependent random variables: Markov chains 2.
• Nov. 22. Martingales: Inequalities, convergence, decompositions.
• Nov. 29. Martingales: Applications.
• Dec. 6. Stationary stochastic processes.
• Dec. 13. 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.
• Practice exam.