Course: MATH 60850, Graduate Probability, Spring 2018
Prerequisite: Math 60350, Real Analysis 1.
Course Content: Review of measure theory, probability spaces, random variables, expected value, independence, laws of large numbers, central limit theorems, random walks, martingales, concentration of measure.
Last update: 2 January 2018

Instructor: Steven Heilman, sheilman(@-symbol)
Office Hours: Mondays, 1PM-2PM, Tuesdays, 10AM-12PM, or by appointment, 118 Hayes-Healy
Lecture Meeting Time/Location: Tuesdays and Thursdays, 1230PM-145PM, DeBartolo Hall 202
Recommended Textbook: Durrett, Probability: Theory and Examples, 4th Edition. (A draft of the book is available online here). I think this is a good book to own if you will study probability and its related fields in the future.
Other Textbooks (not required): I will be drawing on various sources in the course; for example, I will be drawing on some lecture notes of Tao here. These notes complement the Durrett text well.
Dembo's notes available here also complement the Durrett text well.
Feller, An Introduction to Probability Theory and its Applications, Volumes 1 and 2. This set of two books is encyclopedic and very detailed, in contrast to Durrett's intentionally terse book.
Ledoux, The Concentration of Measure Phenomenon. I will include a few results from this book near the end of the course.

Midterm: March 8, 1230PM-145PM
Final Exam: Friday, May 11, 1030AM-1230PM
Other Resources: An introduction to mathematical arguments, Michael Hutchings, An Introduction to Proofs, How to Write Mathematical Arguments
Email Policy:

Exam Procedures: Students must bring their NDID cards to the midterms and to the final exam. Phones must be turned off. Cheating on an exam results in a score of zero on that exam. Exams can be regraded at most 15 days after the date of the exam. This policy extends to homeworks as well. All students are expected to be familiar with the Notre Dame Honor Code. If you are registered with disability services, I would be happy to discuss this at the beginning of the course.
Exam Resources: Here is a page containing a midterm exam and solutions for the 60850 class from Spring 2016. Here is a page containing a final exam and solutions for a similar class. Here is a page containing a final exam for a similar class. Occasionally these exams will cover slightly different material than this class, or the material will be in a slightly different order, but generally, the concepts should be close.

Homework Policy: Grading Policy:

Tentative Schedule: (This schedule may change slightly during the course.)

Week Monday Tuesday Wednesday Thursday Friday
1 Jan 16: Review of measure theory Jan 18: Review of measure theory
2 Jan 23: 1.1, Probability Spaces Jan 25: Homework 1 due. 1.2, Distributions
3 Jan 30: 1.3, Random Variables Feb 1: Homework 2 due. 1.6, Expected Value
4 Feb 6: 1.7, Product measures Feb 8: Homework 3 due. 2.1, Independence
5 Feb 13: 2.2, Weak Law of Large Numbers Feb 15: Homework 4 due. 2.3, Borell-Cantelli Lemmas
6 Feb 20: 2.4, Strong Law of Large Numbers Feb 22: Homework 5 due. 2.4, Strong Law of Large Numbers
7 Feb 27: 3.2, Weak Convergence Mar 1: Homework 6 due. 3.3, Characteristic Functions
8 Mar 6: 3.4, Central Limit Theorems Mar 8: Midterm
9 Mar 13: No class (spring break) Mar 15: No class (spring break)
10 Mar 20: The Lindeberg Replacement Method Mar 22: Homework 7 due. Stein's Method
11 Mar 27: 4.1, Random Walks Mar 29: Homework 8 due. 4.1, Stopping Times
12 Apr 3: 4.2, Recurrence Apr 5: Homework 9 due. 5.1, Conditional Expectation
13 Apr 10: 5.1, Conditional Expectation Apr 12: Homework 10 due. 5.2, Martingales
14 Apr 17: 5.3, Martingale Examples Apr 19: Homework 11 due. 5.4, Doob's Maximal Inequality
15 Apr 24: 5.5, Martingale Convergence Apr 26: Homework 12 due. 5.7, Optional Stopping Theorems
16 May 1: Review of course (last day of class)

Advice on succeeding in a math class:

Homework Homework .tex files Exam Solutions Supplementary Notes