Winter 2017

Instructor: Professor Steve Lalley

Office: 323 Jones Hall

Office Hour: Thursday 1:30 - 2:30

Phone: 702-9890

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

Course Assistant: Si Tang

Office Hour: Fridays, 1:00 - 2:20, Jones 308

This course is the first of a three-quarter sequence in measure-theoretic probability. It is meant for students with a solid grounding in real analysis. The main topics to be covered (not necessarily in the order listed) will be :

- Existence and Extensions of Measures (1 week)
- Independence; Borel's SLLN (1 week)
- Measurable Functions, Transformations, and Random Variables (1 week)
- Integration and Expectation (1-2 weeks)
- Sums of Independent Random Variables (2 weeks)
- Birkhoff's Ergodic Theorem(1 week)
- Weak Convergence (1 week)
- Wigner and Marchenko-Pastur Theorems (1 week)

There will be weekly homework, a midterm and a final exam. Detailed lecture notes will be posted as the course progresses.

- Probability and Measure by Patrick Billingsley
- Probability: Theory and Examples by Richard Durrett

- Assignment 1 Due: Monday January 9
- Assignment 2 Due: Wednesday January 18
- Assignment 3 Due: Monday January 30 Corrected 1/19/2017 2 p.m.
- Assignment 4 Due: Monday February 6
- Assignment 5 Due: Monday February 13
- Assignment 6 Due: Monday February 20
- Assignment 7 Due: Monday February 27
- Assignment 8 Due: Monday March 6 Problem 3(A) corrected3/2/2017 2 p.m.
- Review Exercises

- Measure Theory
- Independence
- Integration and Expectation
- Sums of Independent Random Variables
- Birkhoff's Ergodic Theorem
- Convolutions, Smoothing, and the Central Limit Theorem
- Construction of Brownian Motion