ORF 526: Probability Theory, Fall 2018
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: TuTh 3:00  4:20 pm, 008 Friend Center
Office hours: Th 12:00  2:00 pm, 204 Sherrerd Hall
Teaching Assistants (AIs):

Suqi Liu
Office hours: M 2:00  4:00 pm, 107 Sherrerd Hall (Library room)

Zhuoran Yang
Office hours: W 2:00  4:00 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%
Final info: 3:00 pm, Wednesday, January 16, 2019, 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), 2017. [ online ]
 R. Durrett, Probability: Theory and Examples (4th Edition), 2010. [ 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 Thursday, September 13.
 Lecture 1 (Sep 13): Introduction and overview.

Lecture 2 (Sep 18): Intro to measuretheoretic probability: measurable spaces and measures; Çınlar I.1, I.3
Homework 1 out, due Tuesday, Sep 25  Lecture 3 (Sep 20): Intro to measuretheoretic probability: measures and measurable functions; Çınlar I.2, I.3

Lecture 4 (Sep 25): Intro to measuretheoretic probability: measurable functions and integration; Çınlar I.2, I.4
Homework 2 out, due Tuesday Oct 2  Lecture 5 (Sep 27): Intro to measuretheoretic probability: integration, RadonNikodym theorem, Fubini's theorem, product spaces; Çınlar I.46, II.12

Lecture 6 (Oct 2): Product spaces; weak LLN; Markov, Chebyshev, Chernoff inequalities; convergence in probability and almost sure convergence; Dembo 1.4, 2.1; Çınlar I.6, III.2, III.3, III.6
Homework 3 out, due Tuesday Oct 9  Lecture 7 (Oct 4): BorelCantelli lemmas, strong LLN; Dembo 2.2, 2.3; Çınlar III.2, III.6

Lecture 8 (Oct 9): Strong LLN, CLT, weak convergence; Dembo 2.3, 3.1, 3.2; Çınlar III.5, III.6, III.8
Homework 4 out, due Tuesday Oct 16  Lecture 9 (Oct 11): Weak convergence, Lindeberg's proof of the CLT; Dembo 3.1, 3.2; Çınlar III.5, III.8

Lecture 10 (Oct 16): Characteristic functions; Dembo 3.3; Çınlar II.2, III.5
Homework 5 out, due Friday Oct 26  Lecture 11 (Oct 18): Characteristic functions, CLT, tightness; Markov chains, intro; Dembo 3.3, 6.1, 6.2; Çınlar II.2, III.5, III.8, IV.5
 Lecture 12 (Oct 23): Midterm exam
 Lecture 13 (Oct 25): Markov chains: intro, stationary distribution, classification of states, periodicity; Dembo 6.1, 6.2; Çınlar IV.5

Lecture 14 (Nov 6): Convergence of Markov chains, coupling, stochastic dominance; Dembo 6.1, 6.2; Çınlar IV.5
Homework 6 out, due Tuesday Nov 13  Lecture 15 (Nov 8): Ergodic theorem for Markov chains; Dembo 6.1, 6.2; Çınlar IV.5

Lecture 16 (Nov 13): Markov chains: reversibility, mixing, recurrence, transience; Dembo 6.1, 6.2; Çınlar IV.5
Homework 7 out, due Tuesday Nov 20  Lecture 17 (Nov 15): Conditional expectations; Dembo 4.14.3; Çınlar IV.13
 Lecture 18 (Nov 20): Conditional expectations, martingales; Dembo 4.14.3, 5.1; Çınlar IV.13, V.1V.3

Lecture 19 (Nov 27): Martingales, stopping times, optional stopping theorem; Dembo 5.1, 5.4; Çınlar V.1V.3
Homework 8 out, due Tuesday Dec 4  Lecture 20 (Nov 29): Applications of optional stopping, Pólya urns, martingale convergence; Dembo 5.15.4; Çınlar V.1V.4

Lecture 21 (Dec 4): Martingales: convergence, decomposition, concentration; branching processes; Dembo 5.2, 5.3, 5.5; Çınlar V.1V.4
Homework 9 out, due Tuesday Dec 11  Lecture 22 (Dec 6): Poisson processes, continuous time Markov chains; Dembo 3.4, 8.3.3; Çınlar VI.5

Lecture 23 (Dec 11): Poisson random measures, continuous time stochastic processes, Brownian motion; Dembo 7.17.3; Çınlar VI.2, VIII.1, VIII.7, VIII.8
Homework 10 out, due Monday Jan 14  Lecture 24 (Dec 13): Brownian motion; Dembo 7.3, 9.19.3; Çınlar VIII.1, VIII.7, VIII.8
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