P R O B A B I L I T Y,    F a l l    2 0 2 6

Lectures: Monday, Wednesday, 11:00-12:15pm, in Warren Weaver Hall 512.

Lecturer: Paul Bourgade, office hours Tuesday 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. Send it by email to Yuchen Fan (yf2085@cims.nyu.edu) before 11am every Wednesday.

Grading: oral exam about the homework (25%), midterm (25%) and final (50%).

A tentative schedule for this course is:

• Sep. 2. Measure theory: Introduction, Caratheodory extension theorem (existence).
• Sep. 9. Measure theory: Caratheodory extension theorem (uniqueness), Lebesgue's characterization for ℝ, Integration (random variables).
• Sep. 14. Measure theory: Integration (convergence theorems).
• Sep. 16. Measure theory: Transformations, product spaces.
• Sep. 21. Measure theory: Distributions and expectations.
• Sep. 23. Weak convergence: Characteristic functions, Lévy's theorem
• Sep. 28. Weak convergence: Bochner's theorem.
• Sep. 30. Independent sums: Convolutions, weak law of large numbers.
• Oct 5. Independent sums: Central limit theorem.
• Oct 7. Independent sums: Borel-Cantelli, 0-1 laws.
• Oct 12. Independent sums: Weak and strong law of large numbers.
• Oct 14. Independent sums: Accompanying laws and infinite divisibility.
• Oct. 19. Dependent random variables: Conditioning.
• Oct 21. Dependent random variables: The Radon-Nikodym Theorem.
• Oct. 26. Dependent random variables: Conditional expectation and conditional probability.
• Oct 28. Dependent random variables: Markov chains 1.
• Nov 2. Dependent random variables: Markov chains 2.
• Nov 4. Dependent random variables: Markov chains 3.
• Nov. 9. Dependent random variables: Markov chains 4.
• Nov 11. Dependent random variables: Markov chains 5.
• Nov. 16. Martingales 1.
• Nov 18. Martingales 2.
• Nov. 23. Martingales 3.
• Nov. 25. Martingales 4.
• Nov 30. Martingales 5.
• Dec. 2. Stationary processes 1: ergodic theorems.
• Dec 7 Stationary processes 2: stationary measures.
• Dec. 9. Stationary processes 3: the Markov case.

• Problem set 1. Tex file.