ORF 526: Probability Theory, Fall 2020
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, Zoom
Office hours: W 8:00  9:00 am, W 1:00  2:00 pm, Zoom
Teaching Assistants (AIs):

Suqi Liu
Office hours: M 7:00  8:00 pm, Th 7:00  8:00 pm, Zoom

Jiacheng Zhang
Office hours: Tu 10:00  11:00 am, F 7:00  8:00 pm, Zoom
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 Piazza page (this will be set up when the semester starts).
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.
Think of this as a Q&A wiki for the course, use it for questions and discussions.
Schedule
Classes begin on Monday, August 31.
 Lecture 1 (Aug 31): Introduction and overview.
 Lecture 2 (Sep 2): Intro to measuretheoretic probability: measurable spaces and measures; Çınlar I.1, I.3

Lecture 3 (Sep 7): Intro to measuretheoretic probability: measurable spaces and measures; Çınlar I.1, I.3
Homework 1 out on Sep 8, due Tuesday, Sep 15  Lecture 4 (Sep 9): Intro to measuretheoretic probability: measures and measurable functions; Çınlar I.2, I.3

Lecture 5 (Sep 14): Intro to measuretheoretic probability: measurable functions and integration; Çınlar I.2, I.4
Homework 2 out on Sep 15, due Tuesday, Sep 22  Lecture 6 (Sep 16): Intro to measuretheoretic probability: integration, product spaces, Fubini's theorem; Çınlar I.46, II.12

Lecture 7 (Sep 21): Intro to measuretheoretic probability: integration, product spaces; weak LLN; Markov, Chebyshev, Chernoff inequalities; convergence in probability and almost sure convergence; Çınlar I.46, II.12, III.2, III.3, III.6; Dembo 1.4, 2.1
Homework 3 out on Sep 22, due Tuesday, Sep 29  Lecture 8 (Sep 23): Almost sure convergence, BorelCantelli lemmas, strong LLN; Dembo 2.2, 2.3; Çınlar III.2, III.6

Lecture 9 (Sep 28): Strong LLN, CLT, weak convergence; Dembo 2.3, 3.1, 3.2; Çınlar III.5, III.6, III.8
Homework 4 out on Sep 29, due Tuesday, Oct 6  Lecture 10 (Sep 30): Weak convergence, Lindeberg's proof of the CLT; Dembo 3.1, 3.2; Çınlar III.5, III.8

Lecture 11 (Oct 5): Lindeberg's proof of the CLT, characteristic functions; Dembo 3.1, 3.2, 3.3; Çınlar II.2, III.5, III.8
 Lecture 12 (Oct 7): midterm

Lecture 13 (Oct 14): Characteristic functions, CLT; Dembo 3.1, 3.2, 3.3; Çınlar II.2, III.5, III.8
Homework 5 out on Oct 13, due Tuesday Oct 20 
Lecture 14 (Oct 19): Tightness; Markov chains: intro, stationary distribution; Dembo 3.2, 6.1, 6.2; Çınlar III.5, IV.5
Homework 6 out on Oct 20, due Tuesday Oct 27  Lecture 15 (Oct 21): Markov chains: intro, stationary distribution, classification of states, periodicity; Dembo 6.1, 6.2; Çınlar IV.5

Lecture 16 (Oct 26): Markov chains: stationary distribution, convergence, coupling, stochastic dominance; Dembo 6.1, 6.2; Çınlar IV.5
Homework 7 out on Oct 27, due Tuesday Nov 3  Lecture 17 (Oct 28): Convergence of Markov chains, coupling, ergodic theorem for Markov chains, recurrence, transience; Dembo 6.1, 6.2; Çınlar IV.5
 Lecture 18 (Nov 2): Recurrence and transience of Markov chains, Pólya's theorem; Conditional expectations; Dembo 6.1, 6.2, 4.14.3; Çınlar IV.13, 5
 Lecture 19 (Nov 4): Conditional expectations; Martingales; Dembo 4.14.3, 5.1, 5.4; Çınlar IV.13, V.1V.3

Lecture 20 (Nov 9): Martingales, stopping times; Dembo 5.1, 5.4; Çınlar V.1V.4
Homework 8 out on Nov 10, due Tuesday Nov 17  Lecture 21 (Nov 11): Optional stopping theorem, applications; Dembo 5.1, 5.4; Çınlar V.1V.4

Lecture 22 (Nov 16): Pólya urns, martingale convergence; Dembo 5.15.4; Çınlar V.1V.4
Homework 9 out on Nov 17, due Tuesday Nov 24  Lecture 23 (Nov 18): Doob's decomposition theorem; Poisson processes, superposition, thinning; Dembo 5.2, 5.3, 5.5, 3.4, 8.3.3; Çınlar V.1V.4, VI.2, VI.5
 Lecture 24 (Nov 23): 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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