Course: Math 507A, Probability Theory I, Fall 2020
Prerequisite: MATH 525A or MATH 570
Course Content: Probability spaces; distributions and characteristic functions; laws of large numbers, central limit problems; stable and infinitely divisible laws; conditional distributions.
Syllabus: here. Last update: 12 July 2020

Instructor: Steven Heilman, stevenmheilman(@-symbol)gmail.com
Office Hours: Mondays, 9AM-10AM, 12PM-1PM, on zoom [link posted on blackboard]
Lecture Meeting Time/Location: Mondays, Wednesdays, and Fridays, 1PM-150PM, on zoom [link posted on blackboard]
TA: ..., ...(@-symbol)usc.edu

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.
First Midterm:  Monday, September 21, 1PM-150PM
Second Midterm: Monday, October 26, 1PM-150PM
Final Exam: To be determined.
Other Resources: Supplemental Problems from the textbook. An introduction to mathematical arguments, Michael Hutchings, An Introduction to Proofs, How to Write Mathematical Arguments

Zoom Classroom Conduct: Students should attend zoom lectures in a considerate way and abide by the following rules of decorum. Failure to do so could result in a diminished participation grade. It is preferable (though not required, for equity reasons) that all students have a webcam on during the lecture.
Zoom Security: The zoom links posted on blackboard should not be shared with anyone. You must log into zoom with your USC email address. No one will be admitted to the lecture from the "waiting room" (if you are in the waiting room, you did not log in with your USC email address).
Zoom Technical Support: Technical support for undergraduate students is provided through USC's ITS. Below is the contact information.
Undergraduate Student Technology Support
Portal: https://itsusc.service-now.com/its_sp
Phone: 213-740-5555
Email: consult@usc.edu

Email Policy:

Exam Procedures: If enrollment is ten or lower, midterm exams will be oral exams, where I ask each person questions individually, over zoom (probably for 10-20 minutes). If enrollment goes higher than ten, the midterms will be 24-hour take-home exams, to be submitted on blackboard. In the midterm exams, you are allowed to consult your homeworks, your notes, and your textbook, but these are the only resources you are allowed to use during the exams. So, you are not allowed to use the internet, internet searches, a friend or assistant, etc. Phones must be turned off. If you anticipate issues with a stable internet connection (for obtaining/uploading the exam), issues with obtaining a suitable exam environment, etc., please let me know as soon as possible and we can try to come up with a solution to these issues. 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 USC Student Conduct Code. (See also here.)
Disability Services: If you are registered with disability services, I would be happy to discuss this at the beginning of the course. Any student requesting accommodations based on a disability is required to register with Disability Services and Programs (DSP) each semester. A letter of verification for approved accommodations can be obtained from DSP. Please be sure the letter is delivered to me as early in the semester as possible. DSP is located in 301 STU and is open 8:30am-5:00pm, Monday through Friday.
https://dsp.usc.edu/
213-740-0776 (phone)
213-740-6948 (TDD only)
213-740-8216 (fax)
ability@usc.edu

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Exam Resources: Here are some exams from the last time I taught this class: Exam 1 Exam 1 Solution Final Final Solution. 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 Aug 17: Review of Measure Theory Aug 19: Review of Measure Theory Aug 21: Review of Measure Theory
2 Aug 24: 1.1, Probability Spaces Aug 26: 1.1, Probability Spaces Aug 28: Homework 1 due. 1.2, Distributions
3 Aug 31: 1.3, Random Variables Sep 2: 1.3, Random Variables Sep 4: Homework 2 due. 1.6, Expected Value
4 Sep 7: No class Sep 9: 1.6, Expected Value Sep 11: Homework 3 due. 1.7, Product measures
5 Sep 14: 2.1, Independence Sep 16: 2.2, Weak Law of Large Numbers Sep 18: No homework due. 2.2, Weak Law of Large Numbers
6 Sep 21: Midterm #1 Sep 23: 2.3, Borell-Cantelli Lemmas Sep 25: Homework 4 due. 2.4, Strong Law of Large Numbers
7 Sep 28, 2.4, Strong Law of Large Numbers Sep 30: 3.2, Weak Convergence Oct 2: Homework 5 due. 3.2, Weak Convergence
8 Oct 5: 3.3, Characteristic Functions Oct 7: 3.4, Central Limit Theorem Oct 9: Homework 6 due. 3.4, Central Limit Theorem
9 Oct 12: The Lindeberg Replacement Method Oct 14: Stein's Method Oct 16: Homework 7 due. Stein's Method
10 Oct 19: 4.1, Random Walks Oct 21: 4.1, Random Walks Oct 23: No homework due. 4.1, Stopping Times
11 Oct 26: Midterm #2 Oct 28: 4.2, Recurrence Oct 30: Homework 8 due. 5.1, Conditional Expectation
12 Nov 2: 5.1, Conditional Expectation Nov 4: 5.1, Conditional Expectation Nov 6: Homework 9 due. 5.2, Martingales
13 Nov 9: 5.2, Martingales Nov 11: 5.3, Martingale Examples Nov 13: Homework 10 due. 5.3, Martingale Examples

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Homework

Homework tex files

Supplementary Notes