## MATH 589, WINTER 2019, Advanced Probability Theory 2

### Outline

### Assignments

LaTeX solution template (assignment 1).
LaTeX solution template (assignment 3).
LaTeX solution template (assignment 6).
LaTeX solution template (assignment 7).
Assignment 1.
Assignment 2.
Assignment 3.
Assignment 4.
Assignment 5.
Assignment 6.
Assignment 7.

### Course Notes

### Handwritten notes

- Lecture 1 - Radon-Nikodym, martingales, changes of measure along a filtration, uniform integrability.
- Lecture 2 - Branching process recap, branching processes with immigration, the Kesten-Stigum theorem.
- Lecture 3 - Transforms. Moment generating function, moments of Poisson and Normal random variables, invertibility for non-negative random variables. Discrete Fourier transform. Characteristic functions. Continuity and Inversion theorems.
- Lecture 4 - Weak convergence. Regularity of probability measures, Portmanteau theorem. Tightness, Polish spaces, product spaces. Bounded Lipschitz functions. Skorohod representation theorem, Kolmogorov extension theorem.
- Lecture 5 Exchangeability; de Finetti's theorem; intro to exchangeable arrays.