# MATH 6605 3.00MW (Winter 2015)Probability Theory

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

### Announcements

• The final exam is scheduled Friday May 1, 10am-1pm in N627 Ross.
• Corrected solutions to `Assignment 4 are posted on the assignments page (see below).
• As requested, I will start classes at 10:10am rather than 10:30am, for the rest of the semester.
• Classes resumed on Wednesday March 18, as called for on the York University website. The last class will be on April 17.
• The course meets MWF 10:30, in N814 Ross
• 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

So far we have covered the following topics (sections refer to Rosenthal's book)
• 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)
• Loose ends (Cauchy-Schwarz, Jensen, Fubini)
• Trucation and laws of large numbers (Section 5.4). Here is a cleaned-up proof of the last step of the strong law
• Random series
• Kolmogovor zero-one law (Section 3.5)
• Weak convergence (Chapter 10)
• Characteristic functions and the central limit theorem (Sections 11.1 and 11.2)
At this point we went into survey mode, and I started describing concepts and results, without giving full proofs. You will not be tested on the topics covered in this way, namely:
• Berry-Eseen, Lindeberg-Feller
• law of the iterated logarithm
• stable laws, infinitely divisible laws
• tightness, vague convergence, Helly selection
• extensions of the continuity theorem for characteristic functions, Fourier inversion
• conditional expectations and martingales

### Reserves

The only book on reserve is Rosenthal's A first look at rigorous probability theory. 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.