P R O B A B I L I T Y,    S p r i n g    2 0 1 2

Lectures: Tuesday and Thursday, 1pm-2.30pm, in Science Center 507.

Lecturer: Paul Bourgade, office hours Wednesday 4.30-6pm, you also can email me (bourgade@math.harvard.edu) to set up an appointment or just drop by (Science Center 341).

Course assistant: Constantin Bosinceanu (cbosinceanu@college.harvard.edu). Sections: Tuesday 4-5pm. Office Hours: Sunday at 8pm in the Quincy Dining Hall.

Course description: A rigorous introduction to probability theory, including: discrete and continuous random variables, distribution and density functions in arbitrary dimension, conditional probability, generating functions; weak and strong laws of large numbers, the central limit theorem, random walks, discrete martingales.

Prerequisites: the course will build on infinite series, multivariable calculus, basics about linear algebra, and along the way we will introduce the required notions about set theory and elementary measure theory.

Textbooks: Our reference text will be Probability Essentials, by Jacod-Protter.

Homework: Every Tuesday for the next Tuesday.

Grading: problem sets (50%), midterm (15%) and a final project (35%).

A tentative schedule for this course is:

• Jan. 24. The state space, elements of combinatorics.
• Jan. 26. Random walks.
• Jan. 31. Axioms of probability, σ-algebras.
• Feb. 2. Conditional probability and independence.
• Feb. 7. Random variables on a countable space.
• Feb. 9. Construction of a probability measure on ℝ.
• Feb. 14. Random variables, integration with respect to a probability measure.
• Feb. 16. Independent random variables.
• Feb. 21. Probability distributions on ℝⁿ.
• Feb. 23. Characteristic functions and their properties.
• Feb. 28. Sums of independent random variables.
• March 1. The multivariate Gaussian distribution.
• March 6. Convergence types.
• March 8. Weak convergence and characteristic functions.
• March 20. Laws of large numbers.
• March 22. Central limit theorem: the Lindeberg method.
• March 27. Midterm.
• March 29. Central limit theorem via characteristic functions.
• April 3. L² and Hilbert spaces.
• April 5. Martingales, stopping times.
• April 10. Inequalities for martingales.
• April 12. Convergence of martingales.
• April 17. The Radon-Nikodym theorem.
• April 19. Markov chains I.
• April 24. Markov chains II.

• Problem set 1.
• Problem set 2.
• Problem set 3.
• Problem set 4.
• Problem set 5.
• Problem set 6.
• Midterm exam.
• Problem set 7.
• Problem set 8.