Lectures: Tuesday and Thursday, 2:30-4:00, 2-131

Office hours: Tuesday and Thursday 4:00-5:00 except for colloquium Thursdays Sep. 29, Oct. 6, Nov. 3 and Ahlfors Lecture Thursday October 27. (Happy to stay later on corresponding Tuesdays and/or chat after colloquium.)

Assignments: 7 term problem sets (each worth 10% of grade) and 1 final problem set (worth 30% of grade). Collaboration is encouraged on the first seven problem sets, but students are required to complete the final problem set on their own. (Students are welcome to ask me questions about any of the problem sets during office hours.)

Official course description: Sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales.

Texts: There are many excellent textbooks and sets of lecture notes that cover the material of this course, several written by people right here at MIT. The course material is contained in the union of the following online texts for first-year graduate probability courses:

- S.R.S. Varadhan's lecture notes
- Amir Dembo's lecture notes
- Rick Durrett's book at CiteSeer or at Amazon and here is a recently updated version from Durrett's web page
- Noel Vaillant's www.probability.net tutorials
- Dmitry Panchenko's notes for an earlier rendition of 18.175 .

TENTATIVE SCHEDULE

- Lecture 1 (September 8): Probability spaces, sigma-algebras, extension
- Lecture 2 (September 13): Random variables and distributions
- Lecture 3 (September 15): Integration and expectation
- Lecture 4 (September 20): Laws of large numbers and independence
- Lecture 5 (September 22): Sums of random variables
- Lecture 6 (September 27): Weak laws and moment-generating and characteristic functions PROBLEM SET ONE DUE
- Lecture 7 (September 29): Zero-one laws, maximal inequalities and independent sums
- Lecture 8 (October 4): Large deviations, deMoivre-Laplace and weak convergence
- Lecture 9 (October 6): Large deviations and weak convergence PROBLEM SET TWO DUE
- Lecture 10 (October 13): Characteristic functions and central limit theorem
- Lecture 11 (October 18): Central limit theorem variants and Poisson random variables PROBLEM SET THREE DUE
- Lecture 12 (October 20): Stable random variables
- Lecture 13 (October 25): Higher dimensional limit theorems
- Lecture 14 (October 27): Infinite divisibility and Levy processes PROBLEM SET FOUR DUE
- Lecture 15 (November 1): Random walks
- Lecture 16 (November 3): More random walks
- Lecture 17 (November 8): Reflections and martingales PROBLEM SET FIVE DUE
- Lecture 18 (November 10): More on martingales
- Lecture 19 (November 15): Even more on martingales
- Lecture 20 (November 17): Markov chains PROBLEM SET SIX DUE
- Lecture 21 (November 22): More Markov chains
- Lecture 22 (November 29): Ergodic theory PROBLEM SET SEVEN DUE
- Lecture 23 (December 1): Ergodic theory
- Lecture 24 (December 6): Brownian motion
- Lecture 25 (December 8): More Brownian motion
- Lecture 26 (December 13): More Brownian motion and grand finale FINAL PROBLEM SET EIGHT DUE