Spring 2018

Instructor: Professor Steve Lalley

Office: 323 Jones Hall

Office Hour: Thursday 1:00 - 2:00

Phone: 702-9890

E-mail:
`lalley@galton.uchicago.edu`

Course Assistant: Ahmed Bou-Rabee

E-mail:
`ahmedb@galton.uchicago.edu`
Office Hour: Friday 10:30 - 11:30

Location: Jones 304

This course is the second of a three-quarter sequence in measure-theoretic probability. It is meant for students with a solid grounding in real analysis, including measure and integration, at the level of Stat 381 or Math 312. The main topics to be covered will probably be :

- Hilbert space and the Radon-Nikodym Theorem (1 week)
- Conditional Probability and Expectation (1-2 weeks)
- Discrete-Time Martingales (2 weeks)
- Concentration Inequalities (1 week)
- Coupling and Markov Chains (1-2 weeks)
- Electrical Networks and Reversible Markov Chains (2 weeks)

There will be weekly homework and a final exam.

I will post detailed lecture notes covering all the material that will be discussed during the course, and more. Thus, in principle, you should have no need to buy a textbook. However, if you wish to see standard textbook presentations, you can consult the following books.

- Conditional Expectation and Martingales:
- Probability and Measure by Patrick Billingsley
- Probability with Martingales by David Williams

- Markov Chains:
- Introduction to Stochastic Processes by Greg Lawler

- Electrical Networks and Reversible Markov Chains:
- Random Walks and Electric Networks by P. Doyle and J. L. Snell

- Hilbert Spaces and the Radon-Nikodym Theorem
- Conditional Expectation
- Martingales latest version: April 5 2018
- Martingale Central Limit Theorem latest version: April 5 2018
- Introduction to Concentration Inequalities latest version: April 18 2018
- Random Matrices: Wigner and Marchenko-Pastur Theorems version: May 22, 2019
- Markov Chains: Basic Theory version: May 8, 2018
- Renewal Theorem (Arithmetic Case) version: May 10, 2018
- Electrical Networks and Reversible Markov Chains version: May 30, 2018
- Introduction to Brownian Motion version: May 26, 2019

(More to come later)