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#
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.

### Documents and links

### 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.