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 |

Course material will be based on the book

** Richard Durrett: Probability: Theory and Examples.**

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 |

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. 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 e. 3.4 Central limit theorem for IID sequences with finite variance. ^{it}Homework 4 due. |

8 |
3.4 Lindeberg-Feller theorem. | |

9 |
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10 |
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11 |
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12 |
Thanksgiving break. | |

13 |
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14 |
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15 |

Week | Tuesday | Thursday |
---|---|---|

1 |
1.1-1.7 Probability spaces, random variables. | 1.1-1.7 Expectations, inequalities, types of convergence. |

2 |
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. |

3 |
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. |

4 |
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. |

5 |
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 e. ^{it} |
Midwest Probability Colloquium, Northwestern University, Evanston, IL. Class rescheduled. |

7 |
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. |

8 |
3.7-3.8 Brief discussion about stable and infinitely divisible laws. 3.9 Multivariate normal distribution. CLT in . R^{d} |
Brief discussion about Brownian motion and Donsker's invariance principle. 3.3 Moment problem. |

9 |
4.1 Random walk. Exchangeable sets. Hewitt-Savage 0-1 law. | 4.1 Stopping times. Strong Markov property for random walk. Wald's identity. |

10 |
4.2 Recurrence and transience of simple random walk on . 5.1 Definition of conditional expectation. Z^{d} |
5.1 Properties of conditional expectation. |

11 |
5.1 Conditional probability distributions. | Generalizing Fubini's theorem with stochastic kernels. 5.2 Martingales. |

12 |
5.2 Martingale convergence theorem. | Class rescheduled. |

13 |
5.3 Examples of martingales. | Thanksgiving |

14 |
5.3 Branching process. | Review for exam. Exam on Sunday. |

15 |
Discussion about exam problems. | Return exams. Last passage percolation and KPZ universality. |

- Homework must be handed in by the due date, either in class or by 3 PM either in the instructor's office or mailbox, or as a PDF file uploaded on Canvas. Late submissions cannot be accepted.
- Neatness and clarity are essential. Write one problem per page except in cases of very short problems. Staple you sheets together. You are welcome to use LaTeX to typeset your solutions.
- It is not trivial to learn to write solutions. You have
to write
**enough**to show that you understand the flow of ideas and that you are not jumping to unjustified conclusions, but**not too much**to get lost in details. If you are unsure of the appropriate level of detail to include, you can separate some of the technical details as "Lemmas" and put them at the end of the solution. A good rule of thumb is**if the grader needs to pick up a pencil to check your assertion, you should have proved it.**The grader can deduct points in such cases. - You can use basic facts from analysis and measure theory
in your homework, and the theorems we cover in class
without reproving them. If you find
a helpful theorem or passage in another book,
do not copy the passage but use the
idea to write up your own solution. If you do use other
literature for help, cite your sources
properly. However, it is better to
attack the problems
with your own resources instead of searching the literature
or the internet.
The purpose of the
homework is to strengthen your problem solving skills, not
literature search skills. (
**In particular**: do not use Lyapunov's theorem as a substitute for Lindeberg-Feller unless we have covered Lyapunov's theorem or it has been proved in a homework.) - It is valuable
to discuss ideas for homework problems with other
students. But it is not acceptable to
write solutions together or to copy another person's
solution. In the end you have to hand in your
own
**personal**work. Similarly, finding solutions on the internet is tantamount to cheating. It is the same as copying someone else's solution.