Math 563 - Topics
The nominal text for the course is
Sidney Resnick, A Probability Path (2014 edition)
Available on-line through the U of A library .
The numbers in the following refer to the corresponding chapter
in Resnick
1. Sets and events
Elementary set theory, sigma-algebras,
the sigma-algebra generated by a collection of sets, the
Borel sets.
2. Probability spaces
Definition and elementary properties of a probability measure. Dynkin's
pi-lambda theorem. Construction of probability measures -
extension theorems.
3. Random variables and random elements
Definitions of measurable function, random variables and random elements.
Transformation of a probability measure. Properties of
measurable functions.
4. Independence
Definition of independence of sets, random variables, sigma-algebras.
Zero one laws: Borel-Cantelli lemma, Kolmogorov zero-one law.
5. Integration, expected value
Definition and basic properties of the integral. Limit theorems
for the integral. Transformation theorem. Product spaces,
product measures and Fubini's theorem.
6. Types of convergence
Almost sure convergence, convergence in probability and their
relation. L^p convergence.
7. Laws of large numbers
Weak and strong law of large numbers for an independent,
identically distributed sequence. Convergence of sums of independent
random variables. Glivenko-Cantelli theorem.
8. Convergence in distribution
Distribution function of a random variable. Definition of convergence in
distribution for a random variable and equivalent forms.
Baby Skorohod theorem. Portmanteau theorem (equivalent forms
of convergence of distribution.) Relations between various modes
of convergence.
9. Central limit theorem
Characteristic functions and their relation to convergence in
distribution.
Tightness of families
of probability measures and relation to pre-compactness.
Central limit theorem and more general versions.
10. Martingales
Conditional expectation and martingales.
Note: martingales are studied 565a. This will only be a brief
introduction to them, and we will only cover a fraction of this
chapter.