STAT 4302: Probability for Statistical Inference 2, Winter 2023
Basic info
Course description: This course is the second in a threequarter sequence of graduate probability theory with an eye towards statistical inference.
The first course covers an introduction to mathematical probability theory, including measure theory, law of large numbers, and the central limit theorem. In this course, after a brief review of the aforementioned topics, we cover Markov chains, conditional expectation, martingales, Poisson processes, Brownian motion, and selected advanced topics, together with statistical applications.
Prerequisites: STAT 4301 or permission of instructor.
Instructor: Miklos Z. Racz
Lecture time and location: TuTh 11:00 am  12:20 pm, Tech LG68
Office hours: Th 1  2 pm and 4  5 pm, 2006 Sheridan Rd, Room 108
Teaching Assistant:

Gabriel Pereira Pinto
Office hours: Monday, 3  5 pm, 2006 Sheridan Rd, Basement room
Grading and course policies
Grading: There will be homework problem sets throughout the quarter (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: Wednesday, 15 March, Tech L221
Homework and collaboration policy:
Please be considerate of the grader and write solutions neatly. Unreadable solutions will not be graded.
Please write your name, Northwestern 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.
Resources
We will provide handwritten course notes. In addition, we recommend consulting various textbooks that can aid in your learning. 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:
 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.
Schedule
Tentative schedule and outline of topics covered:
 Lectures 1  2 (Jan 5, 10): Introduction and overview. Review of STAT 4301: measure theory, concentration inequalities, law of large numbers, central limit theorem.
 Lectures 3  7 (Jan 12  26): Markov chains: introduction, stationary distribution, convergence, ergodic theorem, recurrence and transience. Statistical applications.
 Lectures 8  9 (Jan 31, Feb 2): Conditional expectation.
 Lecture 10 (Feb 7): midterm exam.
 Lectures 11  14 (Feb 9  21): Martingales: introduction, stopping times, optional stopping theorem, applications, martingale convergence, and more. Statistical applications.
 Lectures 15  16 (Feb 23, 28): Poisson processes: introduction, superposition, thinning. Continuous time Markov chains. Statistical applications.
 Lectures 17  18 (Mar 2, 7): Brownian motion. Selected topics, time and interest permitting.
 Mar 9: review.
 Mar 15: final exam.
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