Math 280B

Winter 2025, Lecture A00 (Kemp) TuThu 09:30-10:50am

Probability Theory II

Calendar
Syllabus
Homework
Piazza
Gradescope

Course Information

Modality: This is a hybrid, flipped class. Course "lectures" are provided in the YouTube course

@toddkemp-probability ProbabilityTube

Students should watch (and take notes) from the relevant videos corresponding to each "lecture" before the meeting time. The in-person 80-minute meetings will be devoted to active learning: a synopsis of the video lecture content, and then group work, discussion, quizzes, and other interactive tools to promote learning.

Textbook: The main source we will follow are Bruce Driver's excellent Probability notes:

Probability Tools with Examples, by Bruce Driver
- Textbook: The main source we will follow are Bruce Driver's excellent Probability notes:
  
  Probability Tools with Examples, by Bruce Driver
  
  Here are a few other textbooks we recommend as auxiliary sources; all are freely available to UCSD personnel.
  
  Probability: Theory and Examples (5th Edition), by Rick Durrett
  A Probability Path by Sidney Resnick
  A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray
  Probability Theory: A Comprehensive Course by Achim Klenke
  
  In Lectures 28.1, 28.2, 29.1, and 29.2, we discuss Stein's Method. Here are some additional resources for this:
  
  Normal Approximation by Stein's Method by Louis H.Y. Chen, Larry Goldstein, and Qi-Man Shao
  Fundamentals of Stein's Method by Nathan Ross
  Scribe Notes from a 2007 Course in Stein's Method taught by Sourav Chatterjee
  
  If you are not on UCSD campus, make sure you are logged into the VPN in order to gain access; you can find instructions on how to do this here.
- Coursework: There will be weekly homework assignments due on Mondays (starting in Week 2); they will be posted on Gradescope. There will be 3 quizzes, in weeks 4, 7, and 10 of the quarter; they will take place during the last 30 minutes the Tuesday class meeting. There won't be a final exam, but the 10th and final homework set will be due during exam week. Timing and due dates for all courses assessments can be found below.
- Piazza: We will use Piazza, an online discussion board, as a principal communication tool. It will allow you to post messages and answer posts made by your fellow students, about course content, homework, etc. The instructors and TAs will also monitor and post to Piazza regularly. You can access Piazza through Canvas, or directly.
  
  Please note: there are student run forums on Discord and Reddit that may have sections devoted to Math 280B. These are not official course resources. Your instructor may try to interact with students on those forums, mostly to try to make sure any informaiton shared there is accurate, but students should use those, and any other non-official resources, at their own risk. We cannot guarantee that anything posted there is accurate, or conforms to class policies.
- Gradescope: We will use Gradescope to grade your homework, quizzes, and final exam in this course. You will be able to view your graded assignments in Gradescope once they are graded and published.

Instructional Staff

Name	Role	Office	E-mail
Todd Kemp	Instructor	Zoom (coordinates on Canvas)	tkemp@ucsd.edu
Yubo Shuai	Teaching Assistant	HSS 3071	yushuai@ucsd.edu

Our office hours, and all relevant scheduled course activities, can be found in the following calendar.

Calendar

Master List of Meetings and Assessments

All dates and times listed below are Pacific Time.

	Day	Time	Location
Lectures	Asynchronous	Asynchronous	YouTube
Class Meetings	Tuesday, Thursday	09:30-10:50am	APM B402A
Quiz 1	Tuesday, Jan 28	10:20-10:50am	In Class
Quiz 2	Tuesday, Feb 18	10:20-10:50am	In Class
Quiz 3	Tuesday, Mar 11	10:20-10:50am	In Class
Homework 1	Monday, Jan 13	10:00pm	Gradescope
Homework 2	Monday, Jan 20	10:00pm	Gradescope
Homework 3	Monday, Jan 27	10:00pm	Gradescope
Homework 4	Monday, Feb 3	10:00pm	Gradescope
Homework 5	Monday, Feb 10	10:00pm	Gradescope
Homework 6	Monday, Feb 17	10:00pm	Gradescope
Homework 7	Monday, Feb 24	10:00pm	Gradescope
Homework 8	Monday, Mar 3	10:00pm	Gradescope
Homework 9	Monday, Mar 10	10:00pm	Gradescope
Homework 10	Monday, Mar 17	10:00pm	Gradescope

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Recorded Lectures

The Lectures for this course are pre-recorded, and available on YouTube.

The lectures are not divided into even 80-minute chunks. They are organized by topic, concept, or example.

Below, you will find a list (with links) of the lecture videos you should watch prior to the listed date, along with pdf slides of the tablet output during those lecture videos.

Date	Lecture Videos	Slides (Before)	Slides (After)
Jan 7	Cramer's LDP (I)	20.1 Cramer's LDP (I) (Before)	20.1 Cramer's LDP (I) (After)
Jan 7	Cramer's LDP (II)	20.2 Cramer's LDP (II) (Before)	20.2 Cramer's LDP (II) (After)
Jan 9	Total Variation	21.1 Total Variation (Before)	21.1 Total Variation (After)
Jan 9	Law of Rare Events	21.2 Law of Rare Events (Before)	21.2 Law of Rare Events (After)
Jan 14	Weak Convergence	22.1 Weak Convergence (Before)	22.1 Weak Convergence (After)
Jan 14	Weak Convergence over $R^d$	22.2 Weak Convergence over $R^d$ (Before)	22.2 Weak Convergence over $R^d$ (After)
Jan 16	Vague Convergence	23.1 Vague Convergence (Before)	23.1 Vague Convergence (After)
Jan 16	Prokhorov's Compactness Theorem	23.2 Prokhorov's Theorem (Before)	23.2 Prokhorov's Theorem (After)
Jan 16	Skorohod's Theorem	23.3 Skorohod's Theorem (Before)	23.3 Skorohod's Theorem (After)
Jan 21	Complex Integration and Dynkin's Theorem	24.1 Complex Integration (Before)	24.1 Complex Integration (After)
Jan 21	Characteristic Function	24.2 Characteristic Function (Before)	24.2 Characteristic Function (After)
Jan 23	The Riemann-Lebesgue Lemma	25.1 Riemann-Lebesgue (Before)	25.1 Riemann-Lebesgue (After)
Jan 23	Fourier Inversion	25.2 Fourier Inversion (Before)	25.2 Fourier Inversion (After)
Jan 23	The Continuity Theorem	25.3 Continuity Theorem (Before)	25.3 Continuity Theorem (After)
Jan 28	The Central Limit Theorem	26.1 Central Limit Theorem (Before)	26.1 Central Limit Theorem (After)
Jan 28	Infinite Divisibility	26.2 Infinite Divisibility (Before)	26.2 Infinite Divisibility (After)
Jan 30	Average Uniformity	27.1 Average Uniformity (Before)	27.1 Average Uniformity (After)
Jan 30	Lindberg-Feller CLT	27.2 Lindberg-Feller CLT (Before)	27.2 Lindberg-Feller CLT (After)
Jan 30	Random Permutations	27.3 Random Permutations (Before)	27.3 Random Permutations (After)
Feb 4	Probability Metrics	28.1 Probability Metrics (Before)	28.1 Probability Metrics (After)
Feb 4	Stein's Method	28.2 Stein's Method (Before)	28.2 Stein's Method (After)
Feb 6	Berry-Esseen Theorem	29.1 Berry-Esseen (Before)	29.1 Berry-Esseen (After)
Feb 6	Local Dependencies	29.2 Local Dependencies (Before)	29.2 Local Dependencies (After)
Feb 11	Conditional Probability	30.1 Conditional Probability (Before)	30.1 Conditional Probability (After)
Feb 11	Conditional Expectation 1	30.2 Conditional Expectation 1 (Before)	30.2 Conditional Expectation 1 (After)
Feb 13	Orthogonal Projection	31.1 Orthogonal Projection (Before)	31.1 Orthogonal Projection (After)
Feb 13	Conditional Expectation 2	31.2 Conditional Expectation 2 (Before)	31.2 Conditional Expectation 2 (After)
Feb 18	Conditioning as Partial Integration	32.1 Conditioning as Partial Integration (Before)	32.1 Conditioning as Partial Integration (After)
Feb 18	Conditional MCT, Fatou, DCT, Jensen	32.2 Conditional MCT, Fatou, DCT, Jensen (Before)	32.2 Conditional MCT, Fatou, CDCT, Jensen (After)
Feb 18	Conditioning on Random Variables	32.3 Conditioning on RVs (Before)	32.3 Conditioning on RVs (After)
Feb 20	Probability Kernels 1	33.1 Probability Kernels 1 (Before)	33.1 Probability Kernels 1 (After)
Feb 20	Regular Conditional Distributions	33.2 Regular Conditional Distributions (Before)	33.2 Regular Conditional Distributions (After)
Feb 25	Probability Kernels 2	34.1 Probability Kernels 2 (Before)	34.1 Probability Kernels 2 (After)
Feb 25	Random Dynamics	34.2 Random Dynamics (Before)	34.2 Random Dynamics (After)
Feb 27	Stochastic Processes	35.1 Stochastic Processes (Before)	35.1 Stochastic Processes (After)
Feb 27	Poisson Process	35.2 Poisson Process (Before)	35.2 Poisson Process (After)
Mar 4	The Markov Property	36.1 Markov Property (Before)	36.1 Markov Property (After)
Mar 4	Probability Kernels Revisited	36.2 Probability Kernels Revisited (Before)	36.2 Probability Kernels Revisited (After)
Mar 6	Markov Processes	37.1 Markov Processes (Before)	37.1 Markov Processes (After)
Mar 6	Kolmogorov's (Extended) Extension Theorem	37.2 Kolmogorov Extension (Before)	37.2 Kolmogorov Extension (After)
Mar 11	Path Space	38.1 Path Space (Before)	38.1 Path Space (After)
Mar 11	Time Homogeneous Markov Processes	38.2 Time Homogeneous (Before)	38.2 Time Homogeneous (After)
Mar 13	Markov Matrix	39.1 Markov Matrix (Before)	39.1 Markov Matrix (After)
Mar 13	Markov Generator	39.2 Markov Generator (Before)	39.2 Markov Generator (After)

Syllabus

Math 280B is the first quarter of a three-quarter graduate level sequence in the theory of probability. This sequence provides a rigorous treatment of probability theory, using measure theory, and is essential preparation for Mathematics PhD students planning to do research in probability. Math 280A, together with a strong background in undergraduate real analysis at the level of Math 140AB is essential for success in Math 280B; graduate students who do not have this preparation are encouraged instead to consider Math 285, a one-quarter course in stochastic processes. See also this page, maintained by Ruth Williams, for more information on graduate courses in probability at UCSD.

According to the UC San Diego Course Catalog, the topics covered in the full-year sequence 280ABC include the measure-theoretic foundations of probability theory, independence, the Law of Large Numbers, convergence in distribution, the Central Limit Theorem, conditional expectation, martingales, Markov processes, and Brownian motion. The topics for the current version of Math 280A cover everything up to the Law of Large Numbers (with additional topics).

Prerequisite: Students should have mastered the material in Math 280A before enrolling in this class.

Lectures: The lectures for this course are asynchronous, available on YouTube. You should engage with the relevant videos before each "Lecture" session. The scheduled class meetingd times will be devoted to active learning, discussion, and quizzes. These in-class meetings will not be recorded or podcasted. You must attend in-person to gain the benefits of these flipped classes. You must attend quizzes in person for credit.

Homework: Homework assignments are posted on Gradescope, and will be due by 10:00pm on Mondays throughout thee quarter. You must turn in your homework through Gradescope; if you have produced it on paper, you can scan it or simply take clear photos of it to upload. You must select pages corresponding to your solutions of problems during the upload process. Gradescope will allow you to re-select pages at any point until grading has begun. If you have not selected pages when the TA begins grading, the TA will not grade your assignment and you will receive a grade of 0 on it. No appeals of this policy will be considered. It is allowed and even encouraged to discuss homework problems with your classmates and your instructor and TA -- in fact, a significant portion of the in-person class time will be devoted to group work on homework problems. Nevertheless, your final write up of your homework solutions must be your own work.

Quizzes: There will be 3 quizzes throughout the quarter, to test your fundamental knowledge of the course material. You will write them on the Tuesdays of weeks 4, 7, and 10, from 10:20-10:50am, in class. No collaboration (with other humans or with online resources) is allowed on quizzes.

All Scores Count: No quiz or homework grades will be dropped -- all count towards your final grade in the course. If an emergency situation or professional conflict (e.g. travel to a conference to present your research work) conflict with one of the quiz dates, please contact the instructor early to discuss alternatives.

Assessment Versioning: following UCSD (and common) practice, recommended by the Academic Integrity Office, assessments given at non-overlapping times will be comparable (similar length and difficulty, testing the same material), but may not be identical. This practice is meant to maintain course integrity, avoiding unpermitted collaboration (either intentional or accidental).

Here are two links regarding UC San Diego policies on exams:

Exam Responsibilities An outline of the responsibilities of faculty and students with regard to final exams
Policies on Examinations The Academic Senate policy regarding final examinations (These are the rules!)

Regrade Policy: Your quizzes, homeworks, and final exam will be graded using Gradescope. For quizzes and the final exam, you will be able to request regrades through Gradescope for a specified window of time. Be sure to make your request within the specified window of time; no regrade requests will be accepted after the deadline. For homework, any clerical erros (such as a problem or page that the TA accidentally missed when grading) should be discussed with the TA during office hours. Grading rubrics are not negotiable; if the TA has taken off some number of points from your solution, there is a sound pegagogical reason for this. This is a PhD class in mathematics. We are not focused on numerical grades here; we are focused on learning deep and challenging material. The grading is meant as a formative assessment tool; if your grade is not perfect, it indicates you should spend more time reviewing the concepts and thinking about the problems. The TA will give detailed feedback in the grading; it is your responsibility to think and work hard to understand what concepts and ideas you need a firmer understanding of from any assignment where you did not receive full points. Only after working hard on your own, or in collaboration with fellow classmates (for example through Piazza), should you consider approaching your TA or instructor for further explanation of grading choices. However, please understand that these conversations will not result in a change in your grade unless there has been some clear clerical error, such as the TA accidentally missing part of your solution. The TA will not change their assessment of a students work due to conversations or complaints after the fact.

Administrative Deadline: Your scores for all graded work will be posted in Gradescope. It is your responsibility to check your scores and contact your TA before the end of Exam Week to resolve recording errors. Questions regarding missing or incorrectly recorded scores will not be considered after Final Exam Week.

Grading: Your cumulative average will be determined by the following weighting:

70% Homework, 30% Quizzes

We reserve the right to add other optional grading schemes at a later date; if so, your final grade will be computed according to whichever scheme gives you the highest score.

Academic Integrity: In this course, and in your life as a UC San Diego student, we expect you to Excel with Integrity, and to adhere to the UC San Diego Integrity of Scholarship Policy.

Why? Math 280 is a core, foundational course for a wide variety of other mathematics, engineering, and physical science courses. This class is designed to aid your mastery of this important material, for its own sake and for the sake of your learning in all the further courses that rely heavily upon it. Every course component in Math 280 is formulated to cement your understanding, verify what you've mastered, and let us and you know where you need to prioritize your time and energy reviewing. All of our course policies around academic integrity are meant to make sure you are getting the best, most accurate information about your learning in this course. Any students who choose to violate our integrity policies are not just being unfair to their peers; they are ultimately cheating themselves out of a solid foundation in probability theory.

That means we’re all in this together and we actually want the same thing. You, your peers, and the instructional team all want a class that has academic integrity. We want to be able to trust one another, and we want grades to be fair and honest reflections of learning. How can you ensure this type of environment is created in this course? Here are some specific examples:

Do not use, or post to "homework help" services such as Chegg. Do not post solutions to homework, quiz, or exam problems in any public forum (such as Piazza).
Do not collaborate on or copy quizzes or exams. (There will be specific instructions about what that means in an Academic Integrity Pledge at the beginning of each quiz or exam.)
Do not use any apps or sites that do the math for you, or connect you with people who will do the math for you. This will not aid in your learning. This includes any online AI services (e.g.\ ChatGPT and its current incarnations), which should not be used for any work in this class.
Do not attempt to search for solutions to homework, quiz, or exam problems online or in sources outside of the course content. If you accidentally stumble upon a publicly available solution, you should notify your TA and instructor.

We are aware that the temptation to inappropriately collaborate, or use disallowed resources, has been high for several years. We urge you to remember that your integrity is worth more than any advantage you might hope to gain. We will (unfortunately) have to use all tools at our disposal to detect any academic integrity violations, for which there is a zero-tolerance policy. Penalties for these offenses always include assignment of a failing grade in the course, along with administrative sanctions, including up to suspension and dismissal from UC San Diego. Both failing grades and the administrative penalties could impact your eligibility to continue in our graduate programs. Maintain your integrity, and don't risk major consequences to your career at UC San Diego and beyond.

Considerate Conduct: Here are a few of our expectations for etiquette in and out of class.

Entering/exiting class: Please arrive on time and stay for the entire class/section period. If, despite your best efforts, you arrive late, please enter quietly through a rear door and take a seat near where you entered. Similarly, in the rare event that you must leave early (e.g. for a medical appointment), please sit close to a rear exit and leave as unobtrusively as possible.
Noise and common courtesy: When class/section begins, please stop your conversations. Wait until class/section is over before putting your materials away in your backpack, standing up, or talking to friends. Do not disturb others by engaging in disruptive behavior. Disruption interferes with the learning environment and impairs the ability of others to focus, participate, and engage.
Electronic devices: Please do not use devices (such as phones, laptops, and tablets) for non-class-related matters while in class/section. No visual or audio recording is allowed in class/section without prior permission of the instructor/TA.
Messaging etiquette: You are expected to write as you would in any professional correspondence. E-mail, Piazza, and other written communication should be courteous and respectful in manner and tone. Please do not send/post messages that are curt or demanding.

Equity, Diversity, and Inclusion: We are committed to fostering a learning environment for this course that supports a diversity of thoughts, perspectives, and experiences, and respects your identities, including race, ethnicity, heritage, gender, sex, class, sexuality, religion, ability, age, educational background, etc. Our goal is to create a diverse, inclusive, and empowering learning environment where all students feel comfortable and can thrive.

Our instructional staff will make a concerted effort to be welcoming and inclusive to the wide diversity of students in this course. If there is a way we can make you feel more included please let one of the course staff know, either in person, via email/discussion board, or even in a note under the door. Our learning about diverse perspectives and identities is an ongoing process, and we welcome your perspectives and input.

We also expect that you, as a student in this course, will honor and respect your classmates, abiding by the UC San Diego Principles of Community. Please understand that others’ backgrounds, perspectives and experiences may be different than your own, and help us to build an environment where everyone is respected and feels comfortable.

If you experience any sort of harassment or discrimination, please contact the instructor as soon as possible. If you prefer to speak with someone outside of the course, please contact the Office of Prevention of Harassment and Discrimination.

Students with Disabilities: We aim to create an environment in which all students can succeed in this course. If you have a disability, please contact the Office for Students with Disabilities (OSD), which is located in Pepper Canyon Hall, Suite 300, to discuss appropriate accommodations right away. We will work to provide you with the accommodations you need, but you must first provide a current Authorization for Accommodation (AFA) letter issued by the OSD. Students are required to present their AFA letters to faculty (please make arrangements to contact your instructor privately) and to the OSD Liaison in the Math Department (Holly Proudfoot, hproudfood@ucsd.edu) in advance so that accommodations may be arranged. You will find more information here.

Basic Needs and Food Insecurities: If you are experiencing any basic needs insecurities (food, housing, financial resources), there are resources available on campus to help, including The Hub and the Triton Food Pantry. Please visit here to for more information.

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