ORF 526: Probability Theory, Fall 2017
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: W 9:00  11:00 am, 204 Sherrerd Hall
Teaching Assistant (AI): Zongjun Tan
Office hours: F 10:00 am  12:00 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 pm, Thursday, January 18, 2018; location: 006 Friend Center
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
Main resource (required text):
 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 14.
 Lecture 1 (Sep 14): Introduction and overview.

Lecture 2 (Sep 19): Intro to measuretheoretic probability: measurable spaces and measures; Çınlar I.1, I.3
Homework 1 out, due Tuesday, Sep 26  Lecture 3 (Sep 21): Intro to measuretheoretic probability: measurable functions and measures; Çınlar I.2, I.3

Lecture 4 (Sep 26): Intro to measuretheoretic probability: measures and integration; Çınlar I.3, I.4
Homework 2 out, due Tuesday Oct 3  Lecture 5 (Sep 28): Intro to measuretheoretic probability: integration, RadonNikodym theorem, Fubini's theorem, product spaces; Çınlar I.46, II.12

Lecture 6 (Oct 3): Weak LLN; Markov, Chebyshev, Chernoff inequalities; almost sure convergence and BorelCantelli lemmas; Dembo 2.1, 2.2; Çınlar III.2, III.6
Homework 3 out, due Tuesday Oct 10  Lecture 7 (Oct 5): Strong LLN; Dembo 2.3; Çınlar III.6

Lecture 8 (Oct 10): CLT, weak convergence, Lindeberg's proof of the CLT; Dembo 3.1, 3.2; Çınlar III.5, III.8
Homework 4 out, due Tuesday Oct 17  Lecture 9 (Oct 12): Lindeberg's proof of the CLT, characteristic functions; Dembo 3.13.3; Çınlar II.2, III.8

Lecture 10 (Oct 17): Characteristic functions, CLT, tightness; Dembo 3.3; Çınlar II.2, III.5, III.8
Homework 5 out, due Tuesday Oct 24  Lecture 11 (Oct 19): Markov chains, intro; Dembo 6.1, 6.2; Çınlar IV.5
 Lecture 12 (Oct 24): Markov chains, stationary distribution; Dembo 6.1, 6.2; Çınlar IV.5
 Lecture 13 (Oct 26): Midterm exam

Lecture 14 (Nov 7): Convergence of Markov chains; Dembo 6.1, 6.2; Çınlar IV.5
Homework 6 out, due Tuesday Nov 14 
Lecture 15 (Nov 9): Ergodic theorem for Markov chains, recurrence, transience; Dembo 6.1, 6.2; Çınlar IV.5

Lecture 16 (Nov 14): Conditional expectations, martingales; Dembo 4.14.3, 5.1; Çınlar IV.1, V.1, V.2
Homework 7 out, due Tuesday Nov 21  Lecture 17 (Nov 16): Martingales, stopping times, optional stopping theorem; Dembo 5.1, 5.4; Çınlar V.1V.3

Lecture 18 (Nov 21): Applications of optional stopping, Pólya urns, martingale convergence; Dembo 5.15.4; Çınlar V.1V.4

Lecture 19 (Nov 28): cancelled
Homework 8 out, due Tuesday Dec 5  Lecture 20 (Nov 30): cancelled

Lecture 21 (Dec 5): Martingales: convergence, decomposition, concentration; branching processes; Dembo 5.2, 5.3, 5.5; Çınlar V.1V.4
Homework 9 out, due Tuesday Dec 12  Lecture 22 (Dec 7): Poisson processes, continuous time Markov chains; Dembo 3.4, 8.3.3, ; Çınlar VI.5
 Extra Lecture #1 (Dec 8): Guest lecture by Evita Nestoridi on mixing times of Markov chains, with a focus on strong stationary times

Lecture 23 (Dec 12): 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 15  Lecture 24 (Dec 14): Brownian motion; Dembo 7.3, 9.19.3; Çınlar VIII.1, VIII.7, VIII.8
 Extra Lectura #2 (Dec 15): Expository lecture on community detection in the stochastic block model; see this survey and also Abbe's survey
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