**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.

- 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.

- 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.