**Instructor:** David Aldous

**GSI:** Sourav Sarkar

** Class time:** TuTh 12.30 - 2.00 in room 344 Evans.

This is the second half of a year course in mathematical probability at the measure-theoretic level. It is designed for students whose ultimate research will involve rigorous proofs in mathematical probability. It is aimed at Ph.D. students in the Statistics and Mathematics Depts, but is also taken by Ph.D. students in Computer Science, Electrical Engineering, Business and Economics who expect their thesis work to involve probability.

In brief, the course will cover

- Countable state Markov chains. Some old lecture notes. The Diaconis-Freedman Iterated random functions paper.
- (A little) ergodic theory, emphasizing the subadditive ergodic theorem. As background, here is my own elementary introduction to coding and entropy.
- Brownian motion: for more see the book Brownian motion by Peter Morters and Yuval Peres, and the encyclopedic overview by Pitman - Yor.

Week | dates | topics | Durrett sections |
---|---|---|---|

1 | Jan 16 | Recap of Measure theory as used in Probability theory. And link to Tao's three stages. | Chap. 1, Appendix A, sec 2.1.4 |

1 | Jan 18 | Joint distributions correspond to marginal and a kernel. | 5.1.3 |

2 | Jan 23 | Conditional distributions and conditional expectation. The two views of conditional independence. Kolmogorov extension theorem. | 2.1.4, |

2 | Jan 25 | Markov chains: Strong Markov, hitting times and generating function identities. Examples. | 6.1, 6.2 |

3 | Jan 30 | Classification of states; recurrence and transience. | 6.3, 6.4 |

3 | Feb 1 | Invariant measures and stationary distributions. Some old lecture notes | 6.5 |

4 | Feb 6 | Existence of, and convergence to, stationary distributions. | 6.5 |

4 | Feb 8 | Examples of coupling bounds. The Markov chain ergodic theorem. | 6.6 |

5 | Feb 13 | The fundamental matrix. Hitting time formulas via the occupation time identity. Asymptotic variance rates. | Sections 2.1-2.3 of Aldous-Fill |

5 | Feb 15 | Martingale methods for Markov chains. | 6.4 |

6 | Feb 20 | Metropolis algorithm and rejection sampling; general state spaces and Harris chains. | 6.8 |

6 | Feb 22 | Iterated random functions and coupling from the past; continuous-time reversible chains. | See Diaconis-Freedman and section 3.6 of Aldous-Fill. |

7 | Feb 27/ Mar 1 | Overview of weak convergence in metric spaces. Prohorov's theorem and indirect proof via characterization of limit. C[0,1] and D[0,1] and tightness. Examples of convergence of random "objects". | Billingsley Convergence of Probability Measures |

8 | Mar 6/8 | Ergodic theorem; applications to RW. | 7.1, 7.2, 7.3 |

9 | Mar 13/15 | Entropy and Shannon-Breiman-McMillan theorem; subadditive ergodic theorem and applications. | 7.4, 7.5 |

10 | Mar 20/22 | Brownian motion. Existence and path continuity. Invariance properties. Path non-differentiability. Associated martingales and their use in finding distributions, e.g. of hitting time for BM with drift. | 8.1, 8.5 |

Spring break | |||

11 | Apr 3/5 | Reflection principle and formulas derived from it. Mention bridge, excursion, meander. BM as a Gaussian process. Law of iterated logarithm. Skorokhod embedding. | 8.4 |

12 | Apr 10 | Donsker's invariance principle and applications. | 8.6 |

12 | Apr 12 | NO CLASS | |

13 | Apr 17/19 | The three arc sine laws. Martingale central limit theorem via Brownian embedding. Local time and its relevance. | 8.4; Morters-Peres Chap. 6. |

14 | Apr 24 | Levy's theorem. Absolute maximum of Brownian bridge and the Kolmogorov-Smirnov limit. | 8.7; Morters-Peres Chap. 6. |

14 | Apr 26 | de Finetti's theorem and representation of exchangeable arrays. | Tim Austin's Exchangeable random arrays notes. |

15+ | May 3 - 7 | Take-home final exam, given 12.30 Thursday May 3, due 12.30 Monday May 7. |

- STAT 205A - familiarity with measure-theoretic approach to mathematical probability.
- Undergraduate-level familiarity with Markov chains.
- Upper division analysis, e.g. uniform convergence of functions, basics of complex numbers. Basic properties of metric spaces helpful.

There are many other books at roughly the same ``first year graduate" level. Here are my personal comments on some.

**P. Billingsley*** Probability and Measure (3rd Edition)*
Chapters 25-30 make a nice treatment of the "convergence in distribution"
part of 205B.

** J.R. Norris ** *Markov Chains* is a more leisurely account.

**R.M. Dudley*** Real Analysis and Probability*. Best account of
the functional analysis and metric space background relevant for
research in theoretical probability.

**B. Fristedt and L. Gray*** A Modern Approach to Probability Theory*.
700 pages allow coverage of broad range of topics in probability and stochastic
processes.

** R. Bhattacharya and E. C. Waymire **
Stochastic Processes with Applications
gives a broader account of Markov chains, Brownian motion and diffusions, downplaying measure theory.

**L. Breiman*** Probability*.
Classical; concise and broad coverage.

Jim Pitman has his very useful lecture notes linked to the Durrett text (note 3rd edition -- chapter numbers have changed); these notes cover more ground than my course will! Also some lecture notes by Amir Dembo for the Stanford courses equivalent to our 205AB.

**Grading** 60% homework, 40% take-home final.

** Sourav Sarkar ** (souravs@berkeley.edu): Monday 10.00 - 11.00am and Wednesday 4.00 - 5.00pm on room 444 Evans.

If you email us, please put STAT 205B in subject.