Spring 2026: 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 2025 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:
- Advanced central limit theorems
- Conditional expectation
- Martingales
- Markov chains and ergodic theory
- Brownian motion
- Basics of stochastic calculus (time permitting)
Contact & Office Hours
Contact information for the instructor of this course (me) and the teaching assistants is below. The best way to contact us is by email. 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.
| Role | Name | Office Hours | Location | |
|---|---|---|---|---|
| Instructor | Tim Kunisky | kunisky [at] jhu.edu | Wed, 11:00-12:00 | Wyman N445 |
| TA | Xiangyi Zhu | xzhu96 [at] jhu.edu | Thu, 10:30-11:30 | Wyman S425 |
| TA | Ian McPherson | imcpher1 [at] jhu.edu | Fri, 3:00-4:30 | Wyman S425 |
Schedule
Class will meet Mondays and Wednesdays, 1:30pm to 2:45pm in Gilman 400.
Below is a tentative schedule, to be updated as the semester progresses.
| Date | Details |
|---|---|
| Week 1 | |
| Jan 21 | General introduction and logistics. Big-picture review of Probability Theory I: weak convergence, limit theorems, and first methods for proving them. |
| Week 2 | |
| Jan 26 | (Zoom lecture) Lindeberg's exchange principle and alternate proof of the CLT. More robust CLTs. Berry-Esseen quantitative CLT. [Video] [Board PDF] |
| Jan 28 | Moment method for weak convergence. Alternate proofs of central and Poisson limit theorems. |
| Week 3 | |
| Feb 2 | Poisson limit theorems for correlated variables by moment method. Vector limit theorems via Cramér-Wold device. Vector central limit theorem and application to chi-squared test. |
| Feb 4 | Conditional expectation: definition, examples, existence theorem. Statement of Radon-Nikodym theorem. |
| Week 4 | |
| Feb 9 | Proofs of Radon-Nikodym theorem and existence of conditional expectation. Main properties of conditional expectation. |
| Feb 11 | Filtrations and martingales: motivation, definitions, examples. Doob martingale. Martingale transform. The martingale gambling strategy. |
| Week 5 | |
| Feb 16 | Stopping times. Doob's optional stopping theorem. Application to distribution of exit time and exit location in simple random walk. |
| Feb 18 | Doob's upcrossing inequality and almost sure convergence theorem. |
| Week 6 | |
| Feb 23 | (Zoom lecture) Hoeffding concentration inequality for independent sums. Subgaussian random variables. Azuma-Hoeffding concentration inequality for martingales. Bounded differences inequality for nonlinear concentration. Coupon collector / balls-and-bins problem example. [Video] [Board PDF] |
| Feb 25 | (Zoom lecture) Notions of conditioning on events of measure zero: regular conditional probability and disintegration theorem. [Video] [Board PDF] |
| Week 7 | |
| Mar 2 | Uniform integrability. Maximal inequalities for martingales. Martingale convergence theorems in L1 and Lp. Application to random series. |
| Mar 4 | Application of martingale theory to branching processes. Subcritical, critical, and supercritical behavior. Generating function derivation of extinction probability. |
Lecture Notes and Materials
You can find my lecture notes here. The following are books or lecture notes that cover some similar material and might be useful to you in addition to my notes:
- Durrett - Probability: Theory and Examples
- Klenke - Probability Theory: A Comprehensive Course
- Varadhan - Probability Theory
Especially later in the course, we will touch on some rather technical issues about constructions and convergence of continuous-time stochastic processes, for which these might be useful further references:
Assignments & Grading
Grades will be based on written homework assignments (roughly every two weeks) and a take-home final exam. Homework will be posted below, and is to be submitted through Gradescope, typeset in LaTeX.
| Assignment | Assigned | Due | Links |
|---|---|---|---|
| Homework 1 | Jan 30 | Feb 11 | [PDF] |
| Homework 2 | Feb 16 | Mar 2 | [PDF] |
Some further important points about homework:
- Collaboration: You are welcome to discuss homework with your classmates and instructors, but you must write up your own solutions, alone, in your own words. Students found submitting exceedingly similar solutions will not receive any credit for those solutions. If you have discussed the homework with anybody other than instructors, please list their names at the top of your submission.
- Sharing solutions: You may not share full solutions to homework problems with your classmates, show others your written solutions and ask if they are correct, or take notes on or pictures of other students' work before preparing your own solutions.
- AI assistants: You are welcome to use AI assistants (ChatGPT, Gemini, Claude, etc.) to explore the topics discussed in lecture, to clarify any general points of confusion, and to ask broad clarifying questions while doing your homework. You are not allowed to ask them directly for the solutions to homework problems. If you discuss your homework with an AI assistant, describe your interaction at the top of your submission. You are allowed to interact with AI assistants on your homework in the same way you are allowed to with other students: you may discuss problems, but may not ask for or copy complete solutions. If in doubt, you should make sure that you would be able to explain any solution you submit on the blackboard with no other references available.
- Late submissions: You may use a total of five late days (120 hours) 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. Otherwise, after you have used up these late days, further late assignments will be penalized by 20% per day they are late.
- Take-home final exam: None of the above leniencies apply to the exam. It must be completed on time, with absolutely no collaboration or use of AI assistants.