Theory of Probability (MATH230B/STAT310B, Winter 2021)

The second quarter in a yearly sequence of probability theory. Main topics are stopping times, random walks, conditional expectation, discrete time martingales, Markov chains, exchangeability, renewal and ergodic theory.

The spring quarter (Stat310C) is to provide an introduction to continuous time stochastic processes. Specifically, it typically covers continuity and modification, Gaussian and Markov processes, continuous time martingales, Brownian motion and it properties, invariance principles with applications to CLT and LIL, infinitely divisible laws and jump processes.

Prerequisites: Students should be comfortable with integration and measure theory and should have mastered the material of a graduate probability course covering Laws of Large Numbers, Weak Convergence and Central Limit Theorems. Specifically, enrollment is capped and graduate students may register for credit only if they have earned at least a grade A- in Stat310A/Math230A. All other students should email to secure the instructor's permission for doing so (in particular, this applies to all undergraduate/co-term students).

Text: Chapters 4-7 (Chapter 6 only to start of 6.2.1), from STAT310/MATH230 (See also list of changes).

Supplementary texts (most are available online via Stanford):

Meeting: Tu Th 8:30-9:55am (PST). Synchronous, recorded (Panopto edited), white-board (Zoom) lectures, with breakout rooms and TA monitoring of chat-line.

Pre-teaching (non-recorded) Zoom meeting: Th 8:30-9:20am, 1/7 (PST), for informal chat and community building.

Instructor: Amir Dembo, Office hours: Tu 9:55-10:20am, Th 9:55-10:20am (non-recorded, public continuation of lecture Zoom meetings). Use Piazza or TA office hours for all questions (especially on HWs), or e-mail Amir (with MATH230B/STAT310B as your email subject), if a (confidential) private meeting needed (during the term expect reply within 24h).

TA1 (grading HW1, HW3, HW5, HW7, HW9; chat at lectures): Chenyang Zhong. Office hours: Th 11:00-12:30pm (up to 3/15), or e-mail Chenyang (with MATH230B/STAT310B as your email subject).

TA2 (grading HW2, HW4, HW6, HW8, HW9): Yuchen Wu. Office hours: Tu 5:45-7:15pm, We 5:30-7:00pm, Fr 11:00-12:30pm (up to 3/15) or e-mail Yuchen (with MATH230B/STAT310B as your email subject).

Grading : Judgement based on final exam mark (64%), and on consistent Homework (30%) and participation (6%) efforts (details in recording posted on Panopto within Canvas). Typically, total score above 63% needed for passing. Any dispensation for adverse health/housing mandates be contingent on its applicability also for Math. and Stat. PhD-s.

Final Exam: Open books. A 4h timed exam taken by Gradescop within a 24h time frame, starting 5:00am PST, Mon 3/15 (hence upload end-of-frame be 5:00am PST, Tue 3/16).

Material: Chapters 4-6.2 (up to 6.2.1), and Sections 7.1-7.3 of lecture notes.

Study tools: Practice-Exercises (+solutions posted on Canvas) and Practice Final (Posted on Canvsas) with solution via Gradescope, Fri 3/12.

Homework: On a weekly basis, solve four of five homework problems from lecture notes, as posted at HW1-HW9. (+Final Practice Exercises on 2nd page). Problems are due each Saturday 8:00am. Assignments will be submitted through Gradescope by the due date/time. Late homework will not be accepted. Your assignment will typically be graded and returned on Gradescope the following week. Solutions are posted (on the course Canvas page), within 33h of due date.

Syllabus (per lecture notes):

        1/4     Tu(---)              Th(Informal)
	1/11    Tu(4.1.1;4.2)        Th(4.2;4.3)          
	1/18    Tu(4.4;4.1.2)        Th(5.1)          
	1/25    Tu(5.1;5.4)          Th(5.2.2;5.3;5.3.1)
	2/1     Tu(5.2;5.3.1)        Th(5.3.2)     
	2/8     Tu(5.5.1;5.5.2)      Th(5.5.2)
	2/15    Tu(5.5.3)            Th(6.1)     
	2/22    Tu(6.1)              Th(6.1;6.2;7.1)
        3/1     Tu(7.1;7.2)          Th(7.2)
	3/8     Tu(7.2;7.3)          Th(Review)
        3/15    Tu(7.4)              Th(Topics:6.2-6.3)

Material covered, including HWs and self-reading:

See also seminar for current activity in related areas.