Probability Theory (7 1/2 points)
Spring 2018, 3rd reading period


Lecturer and examinator: Jeffrey Steif



Class times and place
To be determined.

Homeworks:

to be put up.

COURSE GOALS:
The aim of the course is that the students learn some of the standard material usually covered in a Ph.D.
course in probability theory. These include, among other things, weak law of large numbers, strong law
of large numbers, random series, characteristic functions (aka Fourier transforms), the central limit
theorem and Brownian motion.

INTENDED AUDIENCE:
This course is intended for graduate students in mathematics, mathematical statistics
as well as masters students with appropriate backgrounds.
(Please feel free to discuss your backround with me to see if it is suitable).
Faculty are also of course very welcome.


PRELIMINARY TOPICS TO BE COVERED:
  • Setting up the language of probability theory in terms of measure and integration theory.
  • Quick review of a.s. convergence, convergence in probability, convergence in L_p and convergence in distribution (aka weak* convergence).
  • General Weak Law of Large Numbers including triangular arrays with examples.
  • General Strong Law of Large Numbers.
  • Random series.
  • Convergence in distribution, characteristic functions (aka Fourier transforms) and the Central Limit Theorem.
  • The Lindeberg-Feller Theorem and some applications.
  • Introduction to Brownian motion and some of its elementary sample path properties.

    Prerequisites:
    Integration theory and having seen some elementary probability theory (feel free to come and
    discuss your background with me).

    Course literature

    There will not be one specific book that the course will be based on and all of the following
    books (for which you can find PDF files for on the internet) are very good books.
    Perhaps the book that will be closest to the lectures is Durrett's.


    Probability: Theory and Examples by Richard Durrett
    A first course in probability theory by Kai Lai Chung
    Probability and Measure by Patrick Billingsley


    Examination Form:
    There will be some homeworks and a final oral exam. For masters students, the final might (or might not) be a written exam.

    Course Examinator:
    Jeff Steif (steif@chalmers.se)

    Registration: Please email Jeff Steif (steif@chalmers.se) for registration. (If you are already on the email list, you do not need to do this.)

    Last modified: Tuesday November 28 1 10:15:34 MET DST 2017