Meetings: TR 12:15, Sterling 1313 
Instructor: Timo Seppäläinen 
Office: 425 Van Vleck. Office Hours: MW 1112 and any other time by appointment 
Office Phone: 2633624 
Email: seppalai at the department's address which is math and dot and wisc and dot and edu 
This is the second semester of the 2semester graduate level introduction to probability theory 733734. The textbook is
Richard Durrett: Probability: Theory and Examples. Version 5 is available from Rick Durrett's homepage at Duke University. The official version for the course is the one dated January 11, 2019.
Material covered. We begin with a quick review of martingales and then take up Sections 4.6 (Uniform integrability) and 4.7 (Backward martingales). The section on backward martingales gives us the opportunity to discuss exchangeable processes and de Finetti's theorem. After that we go on to Markov chains, stationary processes and the ergodic theorem, the subadditive ergodic theorem and some of its applications, and finally Brownian motion and some of its basic properties.
Prerequisites. Some familiarity with key parts from the first semester, such as measuretheoretic foundations of probability, laws of large numbers, central limit theorem, conditional expectations, and martingales.
Course grades will be based on takehome work and possibly one exam.
Week  Tuesday  Thursday 

1 
4.6 Uniform integability, L^{1} convergence.  4.6 L^{1} convergence of martingales, Lévy's 01 law, tail σalgebra, Kolmogorov's 01 law. 4.7 Backward martingales. 
2 
4.7 Exchangeable σalgebra, exchangeable probability measures.  4.7 HewittSavage 01 law, de Finetti's theorem, exchangeable measures as mixtures of IID product measures. 
3 
4.7 de Finetti's theorem as an instance of Choquet's theorem. An application of the HewittSavage 01 law to random walk. 5.15.2 Beginning Markov chains. Markov property of random walk. Homework 1 due. 
5.15.2 Transition probabilities. Construction of Markov chains. 
4 
Introduction to the corner growth model. 5.2 Markov property extended to the infinite future. ChapmanKolmogorov equations. Stopping times and strong Markov property.  5.3 Recurrence, transience and canonical decomposition for discrete state Markov chains. 
5 
5.35.4 Recurrence and transience. 5.5 Invariant and reversible measures and distributions.  5.5 Existence and uniqueness of invariant measures and distributions for countable state space. Stationary processes and shiftinvariant probability measures. 5.6 Periodicity. Homework 2 due. 
6  5.6 Proof of the Markov chain convergence theorem. Dissection principle. Other asymptotic results for Markov chains.  6.1 Stationary processes, measurepreserving dynamical systems, ergodicity. 
7 
6.1 Ergodicity of Markov chains in general state space. Mappings preserve stationarity and ergodicity. Homework 3 due. 
S. Roch lectured on discrete probability. 
8 
6.2 Maximal ergodic lemma and Birkhoff's ergodic theorem.  D. Anderson lectured on Markov Chain Monte Carlo. Homework 4 due. 

SPRING BREAK  SPRING BREAK 
9 
6.4 Subadditive ergodic theorem.  6.4 Conclusion of the proof of the subadditive ergodic theorem. Brief discussion about mixing and trivial tail field. 
10 
Multivariate Gaussian distributions. Lévy's construction of Brownian motion. (B. Valkó lectured.)  7.1 Finitedimensional distributions of Brownian motion and their consistency. Continuous paths do not form a measurable subset of the product space 
11 
7.1 Construction of Brownian motion and Hölder continuity of paths. KolmogorovCentsov criterion for path continuity.  7.1 No Brownian path is Hölder continuous with exponent γ>1/2. 7.2 Markov property. 
12 
7.2 Blumenthal's 01 law, trivial tail field. Homework 5 due. 
7.2 Recurrence of 1dimensional Brownian motion. Brownian filtration. 7.3 Stopping times. 
13 
7.3 Sketch of the proof of the strong Markov property. 7.4 Reflection principle and distribution of the hitting time T_{a}.  7.5 Martingales. 8.1 Discussion of Donsker's theorem. Contrasting weak convergence on C[0,1] and on a countable product space. 
14 
8.1 Skorohod representation. Homework 6 due. 
8.1 Donsker's theorem as a corollary of Skorohod representation. Applications: maximum of random walk, KolmogorovSmirnov statistics for the empirical distribution function. 8.2 Brief discussion about the martingale central limit theorem. 