# MATH 6605 3.00AF (Fall 2017)Probability Theory

Announcements and documents will be posted here as they become available.

### Announcements

• As discussed in class, our final exam will be held Tuesday, December 19. We now have Vari Hall 1018 booked 10am-1pm that day.
• Solutions to Assignment 5 are posted on the assignments page (linked below)
• During exam period, office hours will be: Wednesday Dec 6, 1:30-2:30pm; Tuesday Dec 12, 10-11am; Monday Dec 18, 11-12am, or by appointment.
• The course meets MWF 10:30-11:20 in Vari Hall 3000
• Course reference (optional): A first look at rigorous probability theory by Jeff Rosenthal; 2nd edition, World Scientific. This book is optional - ie you can follow the course purely using notes from class. There are copies in the bookstore. I will keep announce what is coming up, so you can read ahead if you wish. And my approach to this material is quite close to Rosenthal's.

### Topics

We have covered the following topics. I don't assume you've done measure theory, but we will still need to use some results from measure theory. Our approach will be to state results carefully, and give examples from probability, but not to re-prove results that are major topics in a measure theory course. This means the initial material will go by quite fast. Rosenthal's book gives the measure theory proofs, and section references refer to that source.
• Basic measure theory, probability models (Chapters 1-2)
• Random variables and measurable functions (Sections 3.1, 3.3)
• Means and expectations for general random variables (Chapter 4)
• Independence (Section 3.2)
• Weak law of large numbers (Sections 5.1 and 5.3)
• Modes of convergence (Section 5.2)
• Borel Cantelli lemma (Section 3.4)
• Strong law of large numbers (Section 5.3). See SLLN.pdf for a full proof of the strong law.
• Truncation (Section 5.4)
• Convergence of random series
• Kolmogorov zero-one law (Section 3.5)
• Weak convergence (Sections 10.1 and 10.2)
• Introduction to moment generating functions (Section 9.3)
• Characteristic functions, continuity theorem (Section 11.1)
• Central Limit Theorem (Section 11.2)
• Stable laws (proof of stability via characteristic functions.)
• Infinitely divisible laws and the compound Poisson distribution
• Tightness and the full version of the continuity theorem (survey only - you are not responsible for this)
• Conditional expectations and martingales (Chapters 13 and 14).
Reserves The only book on reserve is Rosenthal's A first look at rigorous probability theory, in Steacie Library. The following are not on reserve, but could also be useful:
• A more comprehensive modern text is Durrett's Probability: Theory and Examples.
• Chung's A course in probability theory used to be the standard text for courses like this.
• Lamperti's Probability is a short and conversational book, in much the same spirit as Rosenthal.
• Breiman's Probability is a readable but comprehensive treatment, in the same spirit as Durrett.
• There are parts of the theory that still are best learned from an even older and more encyclopaedic text, namely Feller's An introduction to probability theory and its applications, both volume 1 and volume 2.