## Theory of
Probability, I

Math 451, Spring 2004

The modern understanding of probability
theory is due to Kolmogorov, who in 1933 provided a measure-theoretic
foundation for probability which is now treated as axiomatic. It
provides a means of modelling and analyzing both discrete and continuous
random variables and processes. The goal of this course is
a fairly rigorous understanding of this framework and its basic
implications. The material in this course is fundamental not only
in abstract probabilistic analysis, but also in a number of applied
areas such as communications theory, queueing theory, and mathematical
finance. As an organizing goal, we shall attempt to cover Chapters
1 through 6 of Varadhan's fairly recent book Probability Theory (AMS, 2001).
Namely, we shall cover

- Measure theory

- Weak Convergence

- Independent Sums

- Dependent Random Variables

- Martingales

- Stationary Stochastic Processes (we may not get this far)

Prerequisite: The
material in Math 441 (measure theory) is a prerequisite for this course.

Grading: Grades will
be determined on the basis of homework (30%), a midterm (30%) and a
final (40%).

**Text:**
S. R. S. Varadhan: Probability Theory, American Mathematical
Society, 2001

**Instructor:** Richard Sowers

**Office:** 347 Illini Hall

**Phone:** (217) 333-6246

**email:** r-sowers@math.uiuc.edu

**Home page:** https://math.uiuc.edu/~r-sowers

**Class meets:** Mondays, Wednesdays, and Fridays 3-3:50 P.M. in 347
Altgeld Hall

**Office Hours:** Mondays, Wednesdays, and Fridays 12-12:50 P.M. in
347 Illini Hall