Spring 2025: Probability Theory II (EN.553.721)
Course Description
This is the second semester of a rigorous introduction to measure-theoretic probability for graduate students. I will assume as a prerequisite that you took Probability Theory I (EN.553.720) in Fall 2024 and are comfortable with that material. You are welcome to attend or enroll if you did not, but you will be responsible for working through any material from there that you are not familiar with.
The main goal of this course is to continue to familiarize you with the essential examples of classical probability theory. We will develop a deep understanding of simple random walks, the most foundational such examples. We will also study their generalizations to have dependent steps (martingales), general domains (Markov chains), and continuous paths (Brownian motion). Along the way, we will learn how to formulate and prove probabilistic statements like convergence results, distributional limit theorems, and concentration inequalities, and will introduce many examples of random structures from different areas of mathematics.
The following is a tentative ordered list of the broad topics we will aim to cover:
- Central limit theorem
- Poisson limit theorem and point process
- Conditional expectation
- Martingales
- Markov chains and ergodic theory
- Brownian motion
- Basics of stochastic calculus (time permitting)
Contact & Office Hours
I am the instructor of this course, Tim Kunisky, and the teaching assistant is AMS PhD student Debsurya De.
The best way to contact us is by email, at kunisky [at] jhu.edu and dde4 [at] jhu.edu, respectively. We will decide on a time for office hours in the first week or two of class; in the meantime, please contact us directly to schedule an appointment if you want.
Schedule
Class will meet Mondays and Wednesdays, 1:30pm to 2:45pm in Hodson 211.
Below is a tentative schedule, to be updated as the semester progresses.
| Date | Details |
|---|---|
| Week 1 | |
| Jan 22 | General introduction and logistics. Review of ending of Probability Theory I: modes of convergence, weak convergence, characteristic functions, sketch Lyapunov's proof of Central Limit Theorem (CLT). |
| Week 2 | |
| Jan 27 | Lindeberg's exchange principle and alternate proof of the CLT. Applications of the exchange principle in other universality statements. |
| Jan 29 | Rare events and sparse sums of independent random variables: Poisson limit theorem and Poisson point process. |
| Week 3 | |
| Feb 3 | Poisson point process continued. Examples in extreme value theory and random optimization. |
| Feb 5 | Conditional expectation: intuition, examples, formal definition. |
| Week 4 | |
| Feb 10 | Conditional expectation continued: properties, conditional density and Radon-Nikodym derivative, conditional probability. |
| Feb 12 | Martingales and filtrations: examples and formal definitions. |
| Week 5 | |
| Feb 17 | (Remote lecture / TBD) Optional stopping and convergence theorems for martingales. |
| Feb 19 | (Remote lecture / TBD) Applications of martingales to classical examples: random walks, branching processes. |
Lecture Notes and Materials
I will post handwritten lecture notes shortly after each of our meetings. I will lecture on the blackboard, so you are encouraged to come to all classes if you want to make sure you are following the material in detail. We will mostly follow a combination of the following books:
Grading
Grades will be based on written homework assignments (roughly every two weeks) and a take-home final exam.
Homework will be posted here, and is to be submitted through Gradescope. Further details will be provided before the first assignment. Two important points about homework:
- You are welcome to collaborate with other students on the homework assignments. However, you must write up your own solutions and you may not share your completed solutions with other students. You will receive no credit for a homework assignment if we discover that you have not followed these rules.
- You may use a total of five late days for homework submissions over the course of the semester without penalty. If you need an extension beyond these, you must ask me 48 hours before the due date of the homework and have an excellent reason. After you have used up these late days, further late assignments will be penalized by 20% per day they are late.