|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|
|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. 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.
||3.4 Lindeberg-Feller theorem.|
||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.||2.1 Independence and product measures, independence and convolution, Kolmogorov's extension theorem.|
||2.2 Coupon collector's problem. Weak law of large numbers. St. Petersburg paradox (details left for reading).||2.3 Borel-Cantelli lemmas. Applications to convergence in probability, the necessity of finite mean for the strong law of large numbers, SLLN under a finite fourth moment.|
||2.4 Strong law of large numbers.||2.4 Glivenko-Cantelli theorem. 2.5 Kolmogorov 0-1 law, Kolmogorov's inequality, variance criterion for convergence of random series.|
||3.2 Weak convergence, portmanteau theorem.||3.2 Continuous mapping theorem, Scheffé's theorem, Helly's selection theorem, tightness of sets of probability measures.|
|6 ||3.3 Characteristic function, continuity theorem, an error bound for the Taylor estimate of eit.||Midwest Probability Colloquium, Northwestern University, Evanston, IL. Class rescheduled.|
||3.4 Central limit theorem, Lindeberg-Feller theorem.||3.5 Discussion of the local limit theorem for lattice distributions. 3.6 Poisson limit. Poisson process.|
||3.7-3.8 Brief discussion about stable and infinitely divisible laws. 3.9 Multivariate normal distribution. CLT in Rd.||Brief discussion about Brownian motion and Donsker's invariance principle. 3.3 Moment problem.|
||4.1 Random walk. Exchangeable sets. Hewitt-Savage 0-1 law.||4.1 Stopping times. Strong Markov property for random walk. Wald's identity.|
||4.2 Recurrence and transience of simple random walk on Zd. 5.1 Definition of conditional expectation.||5.1 Properties of conditional expectation.|
||5.1 Conditional probability distributions.||Generalizing Fubini's theorem with stochastic kernels. 5.2 Martingales.|
||5.2 Martingale convergence theorem.||Class rescheduled.|
||5.3 Examples of martingales.||Thanksgiving|
||5.3 Branching process.||Review for exam. Exam on Sunday.|
||Discussion about exam problems.||Return exams. Last passage percolation and KPZ universality.|