Math 230B / Stat 310B : Syllabus

The course has roughly three parts: (1) Conditional expectation; (2) Discrete-time martingales; (3) Markov chains.

Below is a more detailed syllabus (the timeline is only approximate, and the content will be adapted to the needs of the course). Numbers in parentheses refer to Amir Dembo's lecture notes.

Jan 6, 8

Definition of conditional expectation. Existence and uniqueness. Basic properties. Radon-Nykodym theorem. Lebesgue decomposition theorem. Conditional expectation as orthogonal projection in L^2 [4.1, 4.2, 4.3]

Jan 13, 15

Regular conditional probability distributions. Martingales: Definitions, basic examples, stopping times and stopped martingales. Sub- and super-martingales. [4.4, 5.1]

Jan 22

Doob's decomposition, maximal inequalities. [5.2.1, 5.2.2]

Jan 27, 29

Upcrossing inequality Doob's convergence theorem. Convergence of uniformly integrable martingales. Convergence in L^p. Examples and applications [5.3]

Feb 3, 5

Optional stopping theorem. Reverse martingales. Martingale concentration. Branching processes. Exchangeability. [5.4, 5.5]

Feb 10, 12

Markov chains: Definition and canonical construction. Strong Markov property. Examples. [6.1]

Feb 19

Classification of states. Recurrent, transient. [6.2, 6.2.1]

Feb 24, 26

Invariant measures. Existence, uniqueness for irreducible recurrend chains. [6.2.2]

Mar 3, 5

Reversibility, existence of invariant probability measures. Convergence to invariant measure. [6.2.2]

Mar 10, 12

Review.

Homeworks will be assigned every Wednesday, the first time on January 8, and due the following Wednesday (a total of 9 homeworks will be assigned).

An in-class midterm will be given on Friday, February 7. You will be allowed to use the lecture notes.

An in-class final will be given on Thursday, March 20.