UC Davis Math 235B
Probability Theory

Basic information

Topical Outline

-Martingales

- Markov Chains

-Ergodic Theory

Textbook:

Recommended reading:

Lectures

Lecture 1 (January 4): Introduction. Conditional Expectation (section 4.1).

Discussion (January 5): replaced by an office hour since hw problems have yet to be assigned.

Lecture 2 (January 6): Conditional Expectation (section 4.1).

Lecture 3 (January 8): Conditional Expectation (section 4.1), Martingales (section 4.2).

Lecture 4 (January 11): Martingales (section 4.2).

Discussion (January 12).

Lecture 5 (January 13): Martingales. The Martingale Convergence Theorem (section 4.2).

Lecture 6 (January 15): The Martingale Convergence Theorem. Doob's Decomposition (section 4.2).

Discussion (January 19).

Lecture 7 (January 20): Branching Processes (section 4.3).

Lecture 8 (January 22): Branching Processes. Martingales with Bounded Increments (section 4.3).

Lecture 9 (January 25): Kolmogorov's Inequality. Doob's Inequality (section 4.4).

Discussion (January 26).

Lecture 10 (January 27): L^p Maximum Inequality. Convergence in L^p (section 4.4).

Lecture 11 (January 29): Uniform Integrability. Convergence in L^1 (section 4.5).

Lecture 12 (February 1): Convergence in L^1. Levy's 0-1 Law (section 4.5).

Discussion (February 2).

Lecture 13 (February 3): Backwards Martingales (section 4.6).

Lecture 14 (February 5): Hewitt-Savage 0-1 Law. De Finetti's Theorem (section 4.6).

Lecture 15 (February 8): De Finetti's Theorem. Markov Chains (sections 4.6 and 5.1).

Lecture 16 (February 10): Markov Chains. Definitions and Exampoles (section 5.1).

Lecture 17 (February 12): Extensions of Markov Property (section 5.2).

Discussion (February 16).

Lecture 18 (February 17): Extensions of Markov Property (section 5.2).

Lecture 19 (February 19): Review of Midterm Solutions. Extensions of Markov Property (section 5.2).

Lecture 20 (February 22): Reflection Principle (section 5.2).

Discussion (February 23).

Lecture 21 (February 24): Recurrence and Transience (section 5.3).

Lecture 22 (February 26): Recurrence and Transience (section 5.3).

Lecture 23 (March 1): Decomposition Theorem. Examples (section 5.3).

Discussion (March 2).

Lecture 24 (March 3): Birth and Death Chains (section 5.3).

Lecture 25 (March 5): Stationary Measures. Definition and Examples. (section 5.4).

Lecture 26 (March 8): Stationary Measures (section 5.4).

Discussion (March 9).

Lecture 27 (March 10): Stationary Distributions (section 5.4).

Lecture 28 (March 12): Stationary Distributions (section 5.4).

Lecture 29 (March 15): Asymptotic Behavior (section 5.5).

Homework, Midterm, and Final Exam.

Midterm is due in class on Friday, February 12 at 2:10p.m.

Final Exam is due in class on Monday, March 15 at 2:10p.m.

Homework 1

(assigned on January 6):

Homework 1 (PostScript file)

Homework 2

(assigned on January 13):

Homework 2 (PostScript file)

Homework 3

(assigned on January 21):

Homework 3 (PostScript file)

Homework 4

(assigned on January 28):

Homework 4 (PostScript file)

Homework 5

(assigned on February 3):

Homework 5 (PostScript file)

Midterm Exam

Part 2 (Problems 5-8)

Homework 6

(assigned on February 12):

Homework 6 (PostScript file)

Homework 7

(assigned on February 17):

Homework 7 (PostScript file)

Homework 8

(assigned on February 24):

Homework 8 (PostScript file)

Homework 9

(assigned on March 3):

Homework 9 (PostScript file)

Final Exam

Part 2 (Problems 5-8)