Math 571: Advanced Probability — Spring 2026

Instructor: Vladislav Kargin
Office: WH-136
Meeting time and location: TR 8:00–9:30 AM, WH 329
Office hours: TR 10:00–11:00 AM

This course is a 4-credit course, which means that in addition to the scheduled lectures/discussions, students are expected to do at least 9.5 hours of course-related work each week during the semester. This includes things like: completing assigned readings, participating in lab sessions, studying for tests and examinations, preparing written assignments, and other tasks that must be completed to earn credit in the course.

Probability Theory (MATH 501)

This course is an introduction to the advanced concepts of probability theory. It covers topics such as: Measure theory, Probability spaces, Random variables, Conditional Expectations, Stochastic processes, Martingales, Limit Theorems, Large deviations.

The course is intended for students who have a strong foundation in probability theory.

Durrett, Probability: Theory and Examples, 5th edition. PDF available at PTE

Instructor's lecture notes will be provided and posted on Piazza.

We will use Piazza (piazza.com) for communication. All announcements will be sent to the class using Piazza.

Each class session is divided into two parts:

Student-led segment (30–45 minutes): Students take on rotating roles to present and critically examine the day's material.

Lecture segment (45–60 minutes): Instructor extends the material, addresses misconceptions, and covers additional applications.

Each session involves:

• Presenters (2 students): One states definitions, notation, and theorem statements; the other outlines the proof and provides an example.
• Skeptics (2 students): One checks correctness and catches errors; the other proposes counterexamples when assumptions are weakened.
• Scribe (1 student): Records theorem statements, key proof steps, questions raised, and instructor additions. Notes should NOT include names—they are learning material, not meeting minutes. Submit within 24–48 hours; instructor reviews and shares with everyone.
• Observers (3-5 students): Participate in discussion and ask questions; may be called on for examples or perspectives.

• Sunday evening: Instructor announces which pairs are presenters and skeptics for Tuesday and Thursday, and which results will be covered.
• Within-pair role assignment: Students decide among themselves or flip a coin at the start of class.

Students are expected to pre-read the assigned material before each class.

HW 1–3 were graded and count toward the final grade at a reduced weight (see Grading below).

HW 4 onward are ungraded practice problems. Weekly problem sets will continue to be assigned, and detailed solutions will be posted after each set. The problems are not collected or graded, but the material is essential preparation for the midterm, final, and presentations. If you have questions about the problems or want feedback on your proofs, I am happy to discuss them in office hours.

Midterm: One in-class exam (open-book, no internet). Thursday, March 5, 2026.

Final: Take-home exam with a brief (10–12 minutes) oral follow-up. I will choose one of your solutions and ask “why does this step hold?” / “where does the hypothesis matter?” questions. The final is cumulative.

Revised February 22, 2026. See Piazza announcement for details.

~~Original grading policy: Homework 40%, Participation 10%, Midterm 15%, Final write-up 25%, Final oral 10%.~~