MATH 589, WINTER 2019, Advanced Probability Theory 2
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- 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.