Graduate Probability, Spring 2018

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)nd.edu
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:

My email address for this course is sheilman(@-symbol)nd.edu
It is your responsibility to make sure you are receiving emails from sheilman(@-symbol)nd.edu , and they are not being sent to your spam folder.
Do NOT email me with questions that can be answered from the syllabus.

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:

Late homework is not accepted.
If you still want to turn in late homework, then the number of minutes late, divided by ten, will be deducted from the score. (The time estimate is not guaranteed to be accurate.)
The lowest two homework scores will be dropped. This policy is meant to account for illnesses, emergencies, etc.
Do not submit homework via email.
There will be 12 homework assignments, assigned weekly on Thursday and turned in at the beginning of class on the following Thursday.
A random subset of the homework problems will be graded each week. However, it is strongly recommended that you try to complete the entire homework assignment.
You may use whatever resources you want to do the homework, including computers, textbooks, friends, the TA, etc. However, I would discourage any over-reliance on search technology such as Google, since its overuse could degrade your learning experience. By the end of the quarter, you should be able to do the entire homework on your own, without any external help..
All homework assignments must be written by you, i.e. you cannot copy someone else's solution verbatim. However, I would very much encourage you to form study groups and do the homework together in small groups. Homework is the most important part of a graduate mathematics course, and I encourage you to take it very seriously.
Homework solutions will be posted on Friday after the homework is turned in.

Grading Policy:

The final grade is weighted in the following way: homework (40%), midterm (30%), final (30%). Te grade for the semester will be curved. However, anyone who exceeds my expectations in the class by showing A-level performance on the exams and homeworks will receive an A for the class.
If you cannot attend one of the exams, you must notify me within the first two weeks of the start of the quarter. Later requests for rescheduling will most likely be denied.
You must attend the final exam to pass the course.

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:

Review the relevant course material before you come to lecture. Consider reviewing course material a week or two before the semester starts.
When reading mathematics, use a pencil and paper to sketch the calculations that are performed by the author.
Come to class with questions, so you can get more out of the lecture. Also, finish your homework at least two days before it is due, to alleviate deadline stress.
Write a rough draft and a separate final draft for your homework. This procedure will help you catch mistakes. Also, I would very much recommend typesetting your homework. Learning LaTeX is a very important skill to have for doing mathematics. Here is a template .tex file if you want to get started typesetting.
If you are having difficulty with the material or a particular homework problem, review Polya's Problem Solving Strategies, and come to office hours.

Homework

HW1 HW2 HW3 HW4 HW5 HW6 HW7 HW8 HW9 HW10 HW11 HW12 (Solutions Removed)

Homework .tex files

HW1 HW2 HW3 HW4 HW5 HW6 HW7 HW8 HW9 HW10 HW11 HW12

Exam Solutions

Exam 1 Exam 1 Solution Final Final Solution

Supplementary Notes

60850 Lecture Notes