Math 230B / Stat 310B : Syllabus

The course has three main topics 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 9, 11

Definition of conditional expectation. Existence and uniqueness. Basic properties. Radon-Nykodym theorem. Lebesgue decomposition theorem. Conditional expectation as orthogonal projection.

(4.1, 4.2, 4.3)

Jan 16, 18

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

(4.4, 5.1)

Jan 23, 25

Doob's decomposition, maximal inequalities. Upcrossing inequality.

(5.2)

Jan 30, Feb 1

Doob's convergence theorem. Convergence of uniformly integrable martingales. Convergence in L^p. Examples and applications

(5.3, +)

Feb 6, 8

Optional stopping theorem. Reverse martingales. Martingale concentration. Branching processes. Product martingales. Exchangeability.

(5.4, 5.5)

Feb 13, 15

Markov chains: Definition and canonical construction. Strong Markov property. Examples

(6.1)

Feb 20, 22

Countable state space: Classification of states. Recurrence, transience. Invariant measures.

(6.2.1,6.2.2)

Feb 27

Connections between Markov chains and electrical networks (potential theory).

(based on Chapter 2 of this book )

Mar 5, 7

Invariant measures. Ergodic theory: Definitions.

(6.2.3, 6.2.3, 7.1, 7.2)

Mar 12,14

Birkhoff ergodic theorem. Statement of the subadditive ergodic theorem.

Further examples and applications.

Homeworks will be assigned every Thursday. the first time on Jan 11, and due the following Thursday.