# MATH 6605 3.00AF (Fall 2015)Probability Theory

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

### Announcements

• People are welcome to come and look over their final exams in my office, if they are curious about how they did. Information about the distribution of final grades, and final exam grades, is posted on the assignments page.
• The course meets MWF 12:30-1:20. M we're in Chemistry 122, and WF we're in Stong 205
• 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've covered the following topics. The approach with results from measure theory has been to state results carefully, and give examples from probability, but not to re-prove results that are major topics in a measure theory course. Rosenthal's book gives such 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)
• 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 (survey only)
• Infinitely divisible laws (we did the compound Poisson case. The rest was a survey only)
• Tightness and the full version of the continuity theorem (survey only)
• 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.