Instructor: Jing Lei. Office: 132C BH. Email: jinglei [at] andrew [dot] cmu [dot] edu. Office Hours: Wednesday 1-2 or by appointment.

TA: Daren Wang. Office: FMS 326. Email: darenw [at] andrew [dot] cmu [dot] edu, Office hours: Wednesday 4-5 location TBD.

Lecture time and location: TuThu 12-1:20, Wean 4623.

Course description: This is a one-semester course designed to cover two semesters of measure theory and probability. Something will have to give way. Mostly, the proofs will give way. But we will try to prove some of the most important theorems when those proofs illustrate important techniques and concepts. All of the challenging new material in this course will involve limiting operations (epsilon-delta, subsequences, etc).

Prerequisites: Calculus; Real analysis; Undergraduate Probability Theory.

Texts

1. Probability Theory & Measure Theory, 2nd Ed., by R. Ash and C. Dolèans-Dale
2. Probability, Theory and Examples, 4th Ed., by R. Durrett.

Syllabus: A pdf file of the syllabus.

Outline (Lecture notes will be posted approximately weekly before the lectures.)

1. σ-fields, measures, probabilities, extension and uniqueness of probabilities. Lecture Notes Set #1
2. Measurable functions, random variables, distributions, and integration. Lecture Notes Set #2
3. Integration and limits, convergence theorems, Radon-Nikodym derivatives. Lecture Notes Set #3
4. Product spaces and product measures Lecture Notes Set #4
5. Lp space, Weak Law of Large Numbers. Convergence of random variables. Lecture Notes Set #5
6. Almost sure convergence, Strong Law of Large Numbers. Lecture Notes Set #6
7. Convergence in distribution.
8. Characteristic functions and the Central Limit Theorem.
9. Conditional probability and expectation.
10. Martingales.

Homework:

• Homework assignments are posted each Thursday and are due on the following Thursday in class. See the syllabus for detailed homework policies.
• Homework grades will be uploaded on Blackboard by the time the assignments are returned to you. Please report missing grades and grading problems in written form within seven days.
• You are strongly encouraged to submit your homework assignments in pdf format generated by LaTeX. If you are not familiar with LaTeX, here is the tex file for Lecture Notes 1 and you can also find tutorials from google.
1. Homework #1 Due: Jan 26; Last updated: 2017-01-19 1:30 pm [Solutions]
2. Homework #2 Due: Feb 2; Last updated: 2017-01-26 11:20 am [Solutions]
3. Homework #3 Due: Feb 9; Last updated: 2017-02-02 11:45 am [Solutions]
4. Homework #4 Due: Mar 2; Last updated: 2017-02-21 3:55 pm [Solutions]
5. Homework #5 Due: Mar 9; Last updated: 2017-03-02 11:50 am