|Meetings: TR 1-2:15 Ingraham 120|
|Instructor: Timo Seppäläinen|
|Office: 425 Van Vleck. Office hours MW 11-12 or any time by appointment.|
|E-mail: seppalai at math dot wisc dot edu|
Course material will be based on the book
Richard Durrett: Probability: Theory and Examples. (The fourth edition is the newest published one but any edition should work. You can get the book from Rick Durrett's homepage. List of corrections.)
There are numerous good books on probability and it may be helpful to look at other books besides Durrett. For example, these authors have written graduate texts: Patrick Billingsley, Leo Breiman, Kai Lai Chung, Richard M. Dudley, Bert Fristedt and Lawrence Gray, Olav Kallenberg, Sidney Resnick, Albert Shiryaev, Daniel Stroock.
We cover selected portions of Chapters 2-5 of Durrett 4th Ed. These are the main topics:
|Foundations, existence of stochastic processes|
|Independence, 0-1 laws, strong law of large numbers|
|Characteristic functions, weak convergence and the central limit theorem|
Final Exam: Monday 12/18/2017, 2:45PM - 4:45PM, SOC SCI 6203.
|1 ||1.1-1.7 Probability spaces, random variables.|
||1.1-1.7 Expectations, inequalities, types of convergence.||2.1 Independence, proof of the π-λ theorem, application to independence.|
Homework 1 due.
||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.|
||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.|
|| 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.|
||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.
||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.|
|| 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.
||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.
||5.1 Properties of conditional expactation.||5.1 Conditional probability distributions.|
|| 5.1 Generalizing Fubini's theorem with stochastic kernels. 5.2 Martingales.
Homework 6 due.
||5.2 You cannot beat an unfavorable game of chance. Upcrossing lemma.||B. Valkó lectures on random matrices.|
||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.
||Last class of the semester. S. Roch lectures.|