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MATH 6605 3.00MW (Winter 2015)
Announcements and documents will be posted here as they
- 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
- The course meets MWF 10:30, in N814 Ross
- Course reference (optional): A first look at rigorous probability
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
So far we have covered the following topics (sections refer to
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:
- 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
- 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)
- 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,
- conditional expectations and martingales
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.
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.