ORF 526: Probability Theory, Fall 2021
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 Mathematics, Applied & Computational Mathematics, Computer Science, Economics, Electrical and Computer Engineering, and more.
Prerequisites: Undergraduate level probability theory.
Instructor: Miklos Z. Racz
Lecture time and location: MW 11:00 am  12:20 pm, 101 Sherrerd 004 Friend
Office hours: W 12:45  2:45 pm, 204 Sherrerd
Teaching Assistants (AIs):

Daniel Rigobon
Office hours: Tuesday, 10 am  12 pm, 003 Sherrerd

Anirudh Sridhar
Office hours: Thursday, 1  3 pm, 003 Sherrerd
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%
Final info: TBD
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 Ed 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), 2021. [ online ]
 R. Durrett, Probability: Theory and Examples (5th Edition), 2019. [ online ]
 P. Billingsley, Probability and Measure (3rd Edition), 1995.
Think of this as a Q&A wiki for the course, use it for questions and discussions.
Schedule
Classes begin on Wednesday, September 1.
 Lecture 1 (Sep 1): Introduction and overview.
 Lecture 2 (Sep 8): Intro to measuretheoretic probability: measurable spaces and measures; Çınlar I.1, I.3

Lecture 3 (Sep 13): Intro to measuretheoretic probability: measurable spaces and measures; Çınlar I.1, I.3
Homework 1 out on Sep 14, due Tuesday, Sep 21  Lecture 4 (Sep 15): Intro to measuretheoretic probability: measures and measurable functions; Çınlar I.2, I.3

Lecture 5 (Sep 20): Intro to measuretheoretic probability: measurable functions and integration; Çınlar I.2, I.4
Homework 2 out on Sep 21, due Tuesday, Sep 28  Lecture 6 (Sep 22): Intro to measuretheoretic probability: integration, product spaces, Fubini's theorem; Çınlar I.46, II.12

Lecture 7 (Sep 27): Intro to measuretheoretic probability: product spaces; weak LLN; Markov, Chebyshev, Chernoff inequalities; convergence in probability; Çınlar I.46, II.12, III.2, III.3, III.6; Dembo 1.4, 2.1
Homework 3 out on Sep 28, due Tuesday, Oct 5  Lecture 8 (Sep 29): Almost sure convergence, BorelCantelli lemmas, strong LLN; Dembo 2.2, 2.3; Çınlar III.2, III.6

Lecture 9 (Oct 4): Strong LLN; Dembo 2.2, 2.3; Çınlar III.2, III.6
Homework 4 out on Oct 5, due Tuesday, Oct 12  Lecture 10 (Oct 6): Weak convergence, Lindeberg's proof of the CLT; Dembo 3.1, 3.2; Çınlar III.5, III.8

Lecture 11 (Oct 8): Characteristic functions, CLT; Dembo 3.1, 3.2, 3.3; Çınlar II.2, III.5, III.8
 Review (Oct 11): review
 Lecture 12 (Oct 13): midterm

Lecture 13 (Oct 25): Characteristic functions, CLT, tightness; Markov chains: intro; Dembo 3.1, 3.2, 3.3, 6.1, 6.2; Çınlar II.2, III.5, III.8, IV.5
Homework 5 out on Oct 26, due Tuesday Nov 2  Lecture 14 (Oct 27): Markov chains: intro, stationary distribution, classification of states, periodicity; Dembo 6.1, 6.2; Çınlar IV.5

Lecture 15 (Nov 1): Markov chains: stationary distribution, classification of states, periodicity, convergence; Dembo 6.1, 6.2; Çınlar IV.5
Homework 6 out on Nov 2, due Tuesday Nov 9  Lecture 16 (Nov 3): Markov chains: stationary distribution, convergence, coupling, stochastic dominance, ergodic theorem for Markov chains; Dembo 6.1, 6.2; Çınlar IV.5

Lecture 17 (Nov 8): Ergodic theorem for Markov chains, recurrence, transience, Pólya's theorem; Dembo 6.1, 6.2; Çınlar IV.5
Homework 7 out on Nov 9, due Tuesday Nov 16  Lecture 18 (Nov 10): Conditional expectations; Martingales; Dembo 4.14.3; Çınlar IV.13

Lecture 19 (Nov 15): Conditional expectations; Martingales; Dembo 4.14.3, 5.1, 5.4; Çınlar IV.13, V.1V.3
Homework 8 out on Nov 16, due Tuesday Nov 23  Lecture 20 (Nov 17): Martingales, stopping times; Dembo 5.1, 5.4; Çınlar V.1V.4
 Lecture 21 (Nov 22): Optional stopping theorem, applications; Dembo 5.1, 5.4; Çınlar V.1V.4

Lecture 22 (Nov 29): Pólya urns, martingale convergence, Doob's decomposition theorem; Dembo 5.15.5; Çınlar V.1V.4
Homework 9 out on Nov 30, due Tuesday Dec 7  Lecture 23 (Dec 1): Martingale convergence, Doob's decomposition theorem; Poisson processes; Dembo 5.15.5, 3.4, 8.3.3; Çınlar V.1V.4, VI.2, VI.5
 Lecture 24 (Dec 6): Poisson processes, superposition, thinning; Continuous time Markov chains; Poisson random measures; Brownian motion; Dembo 3.4, 8.3.3, 7.3, 9.19.3; Çınlar VI.2, VI.5, VIII.1, VIII.7, VIII.8
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