Miklos Z. Racz

miklos dot racz at northwestern dot edu

• About
• Research
• Students
• Teaching
• Other

ORF 526: Probability Theory, Fall 2019

Basic info

Course description: This is a graduate introduction to probability theory with a focus on stochastic processes.
Topics include: an introduction to mathematical probability theory, law of large numbers, central limit theorem, conditioning, filtrations and stopping times, Markov processes and martingales in discrete and continuous time, Poisson processes, and Brownian motion.

The course is designed for PhD students whose ultimate research will involve rigorous mathematical probability. It is a core course for first year PhD students in ORFE and it is also taken by students in several other areas, such as Applied & Computational Mathematics, Computer Science, Economics, Electrical Engineering, and more.

Prerequisites: Undergraduate level probability theory.

Instructor: Miklos Z. Racz
Lecture time and location: MW 11:00 am - 12:20 pm, 008 Friend Center
Office hours: W 12:45 - 2:45 pm, 204 Sherrerd Hall

Teaching Assistants (AIs):

• Suqi Liu
  Office hours: Th 7:00 - 9:00 pm, 107 Sherrerd Hall (Library room)
  
• Daniel Rigobon
  Office hours: M 3:00 - 5:00 pm, 107 Sherrerd Hall (Library room)
  

---

Grading and course policies

Grading: There will be homework problem sets throughout the semester (approximately weekly), as well as a midterm and a final exam.
Your final score is a combination of your performance in these, with the following breakdown:

• HW 30%
• midterm 30%
• final 40%

Midterm info: Monday, October 21, in class

Final info: 3:00 pm, Wednesday, January 15, 2020, in 006 Friend

Homework and collaboration policy:
Please be considerate of the grader and write solutions neatly. Unreadable solutions will not be graded.
Please write your name, Princeton email, and the names of other students you discussed with on the first page of your HW.
No late homework will be accepted. Your lowest homework score will be dropped.

You should first attempt to solve homework problems on your own.
You are encouraged to discuss any remaining difficulties in study groups of two to four people.
However, you must write up the solutions on your own and you must never read or copy the solutions of other students.
Similarly, you may use books or online resources to help solve homework problems, but you must always credit all such sources in your writeup, and you must never copy material verbatim.

Advice: do the homeworks! While homework is not a major part of the grade, the best way to understand the material is to solve many problems. In particular, the homeworks are designed to help you learn the material along the way.

Email policy: For questions about the material, please come to office hours.
For general interest questions, please post to the course Piazza page.
This facilitates quick and efficient communication with the class.
Please use email only for emergencies and administrative or personal matters.
Please include "ORF 526" in the subject line of any email about the course.

---

Resources

There are many texts that cover first year graduate probability. While the focus and scope of this course is slightly different, these texts can be valuable resources. David Aldous has an extensive annotated list here and here; in particular, consider consulting:

• E. Çınlar, Probability and Stochastics, 2011. [ online ]
• A. Dembo, Lecture notes (for a similar course at Stanford), 2019. [ online ]
• R. Durrett, Probability: Theory and Examples (5th Edition), 2019. [ online ]
• P. Billingsley, Probability and Measure (3rd Edition), 1995.

Piazza: The course has a Piazza page.
Think of this as a Q&A wiki for the course, use it for questions and discussions.

---

Schedule

Classes begin on Wednesday, September 11.

• Lecture 1 (Sep 11): Introduction and overview.
• Lecture 2 (Sep 16): Intro to measure-theoretic probability: measurable spaces and measures; Çınlar I.1, I.3
  Homework 1 out on Sep 17, due Tuesday, Sep 24
• Lecture 3 (Sep 18): Intro to measure-theoretic probability: measures; Çınlar I.3
• Lecture 4 (Sep 23): Intro to measure-theoretic probability: measurable functions and integration; Çınlar I.2, I.4
  Homework 2 out on Sep 24, due Tuesday Oct 1
• Lecture 5 (Sep 25): Intro to measure-theoretic probability: integration, Fubini's theorem, product spaces; Çınlar I.4--6, II.1--2
• Lecture 6 (Sep 30): Intro to measure-theoretic probability: integration, Radon-Nikodym theorem, product spaces; weak LLN; Markov and Chebyshev inequalities; convergence in probability and almost sure convergence; Dembo 1.4, 2.1; Çınlar I.4--6, II.1--2, III.2, III.3, III.6
  Homework 3 out on Oct 1, due Tuesday Oct 8
• Lecture 7 (Oct 2): Chernoff bound, Borel-Cantelli lemmas, strong LLN; Dembo 2.2, 2.3; Çınlar III.2, III.6
• Lecture 8 (Oct 7): Strong LLN, CLT, weak convergence; Dembo 2.3, 3.1, 3.2; Çınlar III.5, III.6, III.8
  Homework 4 out on Oct 8, due Tuesday Oct 15
• Lecture 9 (Oct 9): Lindeberg's proof of the CLT; Dembo 3.1; Çınlar III.5, III.8
• Lecture 10 (Oct 14): Weak convergence, characteristic functions; Dembo 3.2, 3.3; Çınlar II.2, III.5
  Homework 5 out on Oct 15, due Friday Oct 25
• Lecture 11 (Oct 16): Characteristic functions, CLT, tightness; Dembo 3.3; Çınlar II.2, III.5, III.8
• Lecture 12 (Oct 21): Midterm exam
• Lecture 13 (Oct 23): Markov chains: intro, stationary distribution; Dembo 6.1, 6.2; Çınlar IV.5
• Lecture 14 (Nov 4): Markov chains: intro, stationary distribution, classification of states; Dembo 6.1, 6.2; Çınlar IV.5
  Homework 6 out, due Tuesday Nov 12
• Lecture 15 (Nov 6): Markov chains: periodicity, convergence, coupling, stochastic dominance; Dembo 6.1, 6.2; Çınlar IV.5
• Lecture 16 (Nov 11): Convergence of Markov chains, coupling, ergodic theorem for Markov chains; Dembo 6.1, 6.2; Çınlar IV.5
  Homework 7 out, due Tuesday Nov 19
• Lecture 17 (Nov 13): Ergodic theorem for Markov chains, recurrence, transience, Pólya's theorem; Dembo 6.1, 6.2; Çınlar IV.5
• Lecture 18 (Nov 18): Conditional expectations; Dembo 4.1--4.3; Çınlar IV.1--3
  Homework 8 out, due Tuesday Nov 26
• Lecture 19 (Nov 20): Martingales, stopping times; Dembo 5.1, 5.4; Çınlar V.1--V.3
• Lecture 20 (Nov 25): Martingales, stopping times, optional stopping theorem, applications; Dembo 5.1, 5.4; Çınlar V.1--V.4
• Dec 2: cancelled, make-up class on Dec 13
• Lecture 21 (Dec 4): Pólya urns, martingale convergence; Dembo 5.1--5.4; Çınlar V.1--V.4
  Homework 9 out, due Friday Dec 13
• Lecture 22 (Dec 9): Martingales: convergence, decomposition, concentration; Dembo 5.2, 5.3, 5.5; Çınlar V.1--V.4
• Lecture 23 (Dec 11): Poisson processes, Poisson random measures, continuous time Markov chains; Dembo 3.4, 8.3.3; Çınlar VI.2, VI.5
• Lecture 24 (Dec 13): Brownian motion; Dembo 7.3, 9.1--9.3; Çınlar VIII.1, VIII.7, VIII.8

---

Back to Teaching Home