Statistics 383: Measure-Theoretic Probability 2
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.

Recommended Reading:

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

Lecture Notes:

• 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)

Problem Sets:

• Homework 1: Due April 8
• Homework 2: Due April 15
• Homework 3: Due April 22
• Homework 4: Due April 29
• Homework 5: Due May 6
• Homework 6: Due May 13
• Homework 7: Due May 20
• Homework 8: Due June 5