** NEWS:**

**IMPORTANT.**
The best reference, and some of the homeworks, are from
R. Durrett * Probability: Theory and Examples *
**4th Edition**.

**Instructor:** David Aldous

**Teaching Assistant (GSI):**
Wenpin Tang.

** Class time:** TuTh 12.30 - 2.00 in room 70 Evans

** Optional Lab section /GSI Office Hours**
TBA

This is the first 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

- Sketch of pure measure theory (not responsible for proofs)
- Measure-theoretic formulation of probability theory
- Classical theory of sums of independent random variables: laws of large numbers
- Technical topics relating to proofs of above: notions of convergence, a.s. convergence techniques
- Conditional distributions, conditional expectation
- Discrete time martingales
- Introduction to Brownian motion

Week | start | topics | Billingsley | Durrett |
---|---|---|---|---|

1 - 2 | Aug 29, Sep 3/5 | Fields, sigma-fields, measurable functions, measures, Lebesgue measure, distribution functions, coin-tossing, abstract integration | 2,10,13; 3,12,15 | 1.1, 1.2 |

3 | Sep 10/12 | Probability spaces, random variables, expectation, inequalities | 4,5,20 | 1.3 - 1.6 |

4 | Sep 17/19 | Independence, WLLN, Bernstein's theorem, Borel-Cantelli lemmas, 4'th moment SLLN, Glivenko-Cantelli, gambling on favorable game. | 6,20 | 2.1, 2.2 |

5 | Sep 26/28 | a.s. limit theorems for maxima, 2nd moment SLLN, modes of convergence, dominated convergence, maximal inequality, convergence of random series, 1st moment SLLN | 6,21 | 2.3, 2.4 |

6 | Oct 1/3 | variant SLLNs, Fatou, Renewal SLLN. Stopping times, Wald's equation, Kolmogorov 0-1 law; Radon-Nikodym; Cantor measure, decomposition of measures on R. | 22 | 2.4, 2.5 |

7 | Oct 8 | Large deviations | 22, 9 | 4.1, 2.6 |

7/8 | Oct 10/15/17 | joint distributions correspond to marginals and a kernel. Product measure, Fubini's theorem and examples. Kolmogorov consistency theorem. | 32, 33 | 5.1, A4 |

9 | Oct 22/24 | Conditional expectation. Definition and examples of martingales. Convexity. | 18, 34, 36 | 1.7, 5.1 |

10 | Oct 29/31 | Doob decomposition, martingale transforms, stopping times, bounded version of Optional Sampling Theorem. Maximal and upcrossing inequalities. | 35 | 5.2 |

11 | Nov 5/7 | MG convergence theorems, Levy 0-1 law, L_p convergence, conditional Borel-Cantelli, Kakutani's theorem, general form of optional sampling, MG analog of Wald. | 35 | 5.3, 5.7 |

12 | Nov 12/14 | Boundary crossing examples. Galton-Watson processes. Patterns in coin-tossing. MG proof of Radon-Nikodym theorem. Azuma's inequality; examples. | 35 | 5.3, 5.4, 5.5 |

13 | Nov 19 | Reversed MGs and SLLN. Exchangeability and de Finetti's theorem. Kolmogorov consistency theorem. | 35 | 5.3, 5.7 |

13-15 | Nov 21/26, Dec 3/5 | 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. Reflection principle and formulas derived from it. Distribution of last zero before/first zero after t=1. Mention bridge, excursion, meander. | 37 | 8.1 - 8.5 |

Take-home final. 2.00pm Thursday 12/5 -- 2.00pm Monday 12/9 |

- Upper division probability - familiarity with calculations using random variables.
- Upper division analysis, e.g. uniform convergence of functions, basics of complex numbers. Basic properties of metric spaces helpful.

**P. Billingsley*** Probability and Measure (3rd Edition)*
Chapters 1-30 contain a more careful and detailed treatment of the
topics of this semester, in particular the measure-theory background.
Recommended for students who have not done measure theory.

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

**D. Khoshnevisan** * Probability * is a well-written concise account of the key
topics in 205AB.

** K.L. Chung*** A Course in Probability Theory* covers
many of the topics of 205A: more leisurely than Durrett and
more focused than Billingsley.

** D. Williams** *Probability with Martingales * has a uniquely enthusiastic style;
concise treatment emphasizes usefulness of martingales.

**Y.S. Chow and H. Teicher*** Probability Theory*.
Uninspired exposition, but has useful variations on technical topics such as
inequalities for sums and for martingales.

**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.

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

**O. Kallenberg** * Foundations of Modern Probability*.
Quoting an amazon.com reviewer: ``.... a compendium of all the relevant results of probability
..... similar in breadth and depth to Loeve's classical text of the mid 70's.
It is not suited as a textbook, as it lacks the many examples that are needed to absorb the
theory at a first pass. It works best as a reference book or a "second pass" textbook."

**John B. Walsh** *Knowing the Odds: An Introduction to Probability*.
New in 2012. Looks very nice -- concise treatment with quite challenging exercises developing part of theory.

** Santosh Venkatesh**
* The Theory of Probability: Explorations and Applications*.
Unique new book, intertwining a broad range of undergraduate and graduate-level topics for an applied audience.

Jim Pitman has his very useful lecture notes linked to the Durrett text; 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% final.

**Wenpin Tang** (x09@berkeley.edu)
Monday 3.00 - 4.00pm in room 340 Evans and
Friday 11.15 - 12.15 in 444 Evans.

**if you email us** put "STAT 205A" in subject.