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 ℝ
^{n}. - 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. L
^{2}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.