18.175 Theory of Probability: Fall, 2016

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 .

A gentler introduction to some of the material in the course appears in David Gamarnik's notes. For a more general analysis reference, there is also the online text Applied Analysis by Hunter and Nachtergaele.

Other excellent graduate probability books (that I don't think have been posted online, at least not by the authors) include (but are obviously not limited to) Patrick Billingsley's book , Richard Dudley's book , Dan Stroock's book and David Williams' book. There's a lot of overlap between these books, but you'll develop strong opinions if you spend much time with them. Here is one person's rated list of graduate probability books. (You probably won't agree with the list author's opinions, but it's still a nice list.) This course will more or less follow Durrett's treatment, with supplemental material and perspectives in lectures. Most of the material in the slides is lifted pretty directly from Durrett or from one of the other online sources listed above. I use the same notation as Durrett whenever possible.

Stellar course web site

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