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
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
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
also complement the Durrett text well.
Feller, An Introduction to
and its Applications, Volumes 1 and 2. This set of two books is
encyclopedic and very detailed, in contrast to Durrett's intentionally
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
introduction to mathematical
arguments, Michael Hutchings,
An Introduction to Proofs,
How to Write Mathematical Arguments
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.
- My email address for this course is sheilman(@-symbol)nd.edu
- It is your responsibility to make sure you are receiving emails from
, and they are not being sent to your spam folder.
- Do NOT email me with questions that can be answered from the syllabus.
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.
is a page containing a final exam for a similar class.
Occasionally these exams will
slightly different material than this class, or the material will be in a slightly
different order, but generally, the concepts should be close.
- 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
in at the beginning of class on the following
- 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 |
||Jan 16: Review of measure theory
||Jan 18: Review of measure theory
||Jan 23: 1.1, Probability Spaces
||Jan 25: Homework 1 due. 1.2, Distributions
||Jan 30: 1.3, Random Variables
||Feb 1: Homework 2 due. 1.6, Expected Value
||Feb 6: 1.7, Product measures
||Feb 8: Homework 3 due. 2.1, Independence
||Feb 13: 2.2, Weak Law of Large Numbers
||Feb 15: Homework 4 due. 2.3, Borell-Cantelli Lemmas
||Feb 20: 2.4, Strong Law of Large Numbers
||Feb 22: Homework 5 due. 2.4, Strong Law of Large Numbers
||Feb 27: 3.2, Weak Convergence
||Mar 1: Homework 6 due. 3.3, Characteristic Functions
||Mar 6: 3.4, Central Limit Theorems
||Mar 8: Midterm
||Mar 13: No class (spring break)
||Mar 15: No class (spring break)
||Mar 20: The Lindeberg Replacement Method
||Mar 22: Homework 7 due. Stein's Method
||Mar 27: 4.1, Random Walks
||Mar 29: Homework 8 due. 4.1, Stopping Times
||Apr 3: 4.2, Recurrence
||Apr 5: Homework 9 due. 5.1, Conditional Expectation
||Apr 10: 5.1, Conditional Expectation
||Apr 12: Homework 10 due. 5.2, Martingales
||Apr 17: 5.3, Martingale Examples
||Apr 19: Homework 11 due. 5.4, Doob's Maximal Inequality
||Apr 24: 5.5, Martingale Convergence
||Apr 26: Homework 12 due. 5.7, Optional Stopping Theorems
||May 1: Review of course (last day of class)
Advice on succeeding in a math class:
Homework .tex files
- Review the relevant course material before you come to lecture.
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
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
Strategies, and come to office hours.