Meetings: TR 1-2:15, Sterling 1313 |

Instructor: Timo Seppäläinen |

Office: 425 Van Vleck. Office Hours: MW 11-12 and any other time by appointment |

Office Phone: 263-3624 |

E-mail: seppalai at the department's address which is math and dot and wisc and dot and edu |

This is the second semester of the 2-semester graduate level introduction to probability theory 733-734. The textbook is

** Richard Durrett: Probability: Theory and Examples. ** Version 5 is available from Rick Durrett's homepage at Duke University.

**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 measure-theoretic foundations of probability,
laws of large numbers, central limit theorem, conditional expectations, and
martingales.

Course grades will be based on take-home work and possibly one exam.

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

1 |
Review of conditional expectation and martingales. 5.4 Doob's inequality, L convergence of martingales. ^{p} |
5.5 Uniform integability, L convergence of martingales. ^{1} |

2 |
Applications of L convergence (S. Roch). ^{1} |
Optional stopping (S. Roch). |

3 |
5.6 Backward martingales. | 5.6 Exchangeable processes and deFinetti's theorem. |

4 |
Class rescheduled. | Beginning Markov chains (B. Valkó). |

5 |
Discussion: Extreme points, exchangeable measures, and Choquet's theorem. Martingale CLT. Burkholder-Davis-Gundy inequalities. | 6.1-6.2 Markov chains. |

6 | 6.3 Strong Markov property. | 6.3 Reflection principle. 6.4 Recurrence and transience for countable state space. |

7 |
6.4 Recurrence and transience for countable state space. | 6.4 Martingale argument for recurrence. |

8 |
6.5 Invariant distributions and measures. | 6.5 Invariant distributions and measures. |

9 |
6.6 Convergence theorem. Final comments on Markov chains: law of large numbers, recurrence and superharmonic functions. | 7.1 Stationary processes, measure-preserving dynamical systems, ergodicity. |

10 |
7.1 Ergodicity of an irreducible Markov chain and an irrational rotation of the circle. 7.2 Maximal ergodic lemma. | 7.2 Proof of Birkhoff's ergodic theorem. |

11 |
Ergodicity, mixing and trivial tail field. Markov chain CLT via martingale CLT and the ergodic theorem. | Solving Poisson's equation for a Markov chain. 7.4 Examples of subadditive ergodicity. |

12 |
7.4 Kingman-Liggett subadditive ergodic theorem: proof of almost sure convergence. | 7.4 Kingman-Liggett subadditive ergodic theorem: proof of L convergence. Applications. ^{1} |

13 |
8.1 Brownian motion: definition, the Gaussian (heat) kernel, discussion about the construction of Brownian motion, Kolmogorov's criterion for path continuity. | 8.1 Proof of Kolmogorov's criterion, completion of the construction of Brownian motion. |

14 |
Lévy's construction of Brownian motion. Nondifferentiability of paths. Quadratic variation. (B. Valkó). | 8.2 Markov property of Brownian motion, Blumenthal's 0-1 law, trivial tail field, Brownian filtrations. (B. Valkó). |

15 |
8.3 Strong Markov property of Brownian motion. Hitting times. | 8.6 Donsker's theorem. |

Timo Seppalainen