Topics

- Sections: 7.7, 7.8, 7.9
- Definitions:
- Conditional Expectation with respect to a sigma field
- Sigma field generated by a set of RV's
- Measurable with respect to a sigma field
- A filtration of sigma fields
- Martingale
- Properties of Conditional Expectation
- Martingale Converge Theorem
- Azuma's Inequality
- Isoperimetric Inequalities

- Theorems: 7.7.10, 7.7.11, 7.8.1, 7.9.24, 7.9.26
- Sample Questions:
- Prove that S_n is a martingale
- Prove that S_n^2 - n is a martingale
- Prove that Z_n/b^n is a martingale where Z_n is a branching process and b is the mean offspring size
- State and prove Azuma's Inequality
- Let S_0 =0, S_1, S_2,... be a martingale. What is E (S_n) ?