Meetings: TR 1-2:15 Ingraham 120 |
Instructor: Timo Seppäläinen |
Office: 425 Van Vleck. Office hours MW 11-12 or any time by appointment. |
Phone: 263-3624 |
E-mail: seppalai at math dot wisc dot edu |
Foundations, existence of stochastic processes |
Independence, 0-1 laws, strong law of large numbers |
Characteristic functions, weak convergence and the central limit theorem |
Random walk |
Conditional expectations |
Martingales |
Week | Tuesday | Thursday |
---|---|---|
1 | 1.1-1.7 Probability spaces, random variables. | |
2 | 1.1-1.7 Expectations, inequalities, types of convergence. | 2.1 Independence, proof of the π-λ theorem, application to independence. Homework 1 due. |
3 | 2.1 Independence and product measures, independence and convolution, Kolmogorov's extension theorem. | 2.3 Borel-Cantelli lemmas. Applications to convergence in probability, the necessity of finite mean for the strong law of large numbers. |
4 | 2.4 Strong law of large numbers. Homework 2 due. | 2.4 Glivenko-Cantelli theorem. 2.5 Tail σ-algebra, Kolmogorov 0-1 law, Kolmogorov's inequality. |
5 | 2.5 Variance criterion for convergence of random series. Separate lecture notes: The corner growth model, its queueing interpretation, superadditivity, the exactly solvable case with exponentially distributed weights. | 3.2 Weak convergence, portmanteau theorem. |
6 | 3.2 Continuous mapping theorem, Scheffé's theorem, Helly's selection theorem, tightness of sets of probability measures. Homework 3 due. | Midwest Probability Colloquium at Northwestern University. Class rescheduled. |
7 | 3.2 Completion of tightness discussion. 3.3 Characteristic functions. Continuity theorem. | 3.3 Completion of the proof of the continuity theorem. An error bound for the Taylor estimate of eit. 3.4 Central limit theorem for IID sequences with finite variance. Homework 4 due. |
8 | 3.3, 3.5 Discussion of the Berry-Esseen theorem and the local limit theorem. 3.4 Lindeberg-Feller theorem. | 3.6 Poisson limit. Poisson process. 3.7-3.8 Brief discussion about stable and infinitely divisible laws. 3.9 Weak convergence in Rd. |
9 | 3.9 Multivariate normal distribution. CLT in Rd. Separate lecture notes: Brief discussion of the Tracy-Widom distribution and the fluctuations of the corner growth model. | 4.1 Random walk. Exchangeable sets. Hewitt-Savage 0-1 law. Homework 5 due. |
10 | 4.1 Stopping times. Strong Markov property for random walk. Wald's identity. Gambler's ruin. | 4.2 Recurrence and transience of simple random walk on Zd. 5.1 Conditional expectation. |
11 | 5.1 Properties of conditional expactation. | 5.1 Conditional probability distributions. |
12 | 5.1 Generalizing Fubini's theorem with stochastic kernels. 5.2 Martingales. Homework 6 due. | Thanksgiving break. |
13 | 5.2 You cannot beat an unfavorable game of chance. Upcrossing lemma. | B. Valkó lectures on random matrices. |
14 | 5.2 Martingale convergence theorem. Random walk example of failure of L1 convergence. 5.3 Pólya's urn. Galton-Watson branching process: extinction when μ<1. | 5.4 Doob's inequality. Lp convergence of martingales for p>1. 5.5 Definition of uniform integrability. Homework 7 due. |
15 | Last class of the semester. S. Roch lectures. |