Martingales

MATH 4221

Fall 2013

Skiles 254

Instructor

Will Perkins

wperkins3@math.gatech.edu

Office: Skiles 017

Topics
  • Sections: 7.7, 7.8, 7.9
  • Definitions:
    1. Conditional Expectation with respect to a sigma field
    2. Sigma field generated by a set of RV's
    3. Measurable with respect to a sigma field
    4. A filtration of sigma fields
    5. Martingale
    6. Properties of Conditional Expectation
    7. Martingale Converge Theorem
    8. Azuma's Inequality
    9. Isoperimetric Inequalities
  • Theorems: 7.7.10, 7.7.11, 7.8.1, 7.9.24, 7.9.26
  • Sample Questions:
    1. Prove that S_n is a martingale
    2. Prove that S_n^2 - n is a martingale
    3. Prove that Z_n/b^n is a martingale where Z_n is a branching process and b is the mean offspring size
    4. State and prove Azuma's Inequality
    5. Let S_0 =0, S_1, S_2,... be a martingale. What is E (S_n) ?