Theory of Probability I

**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`(this syllabus can be found there)**Office Hours:**10:30-11:30 MWF and by appointment**Class meets:**MWF 1-1:50 P.M. in 447 Altgeld**Text:**My Notes

This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern nprobability theory (see also the Course Catalog Entry for Math 451 or the list of UIUC Probability Theory Classes). While I do not wish to sound discouraging, this will be an honest course. When you finish, you will feel comfortable with the ``plumbing'' of modern probability.

- Introduction and probabilistic framework
- Connections with measure theory
- Convergence of random variables
- a.s.
- convergence in L-p
- convergence in probability
- convergence in distribution

- Information, Independence, and Conditioning
- Sigma-algebra manipulation
- The calculus of sigma-algebras
- Stopping times
- Independence
- Conditioning
- Asymptotics: limit theorems
- The weak law of large numbers for square-integrable random variables
- Large deviations
- Central Limit Theorem
- Martingales
- Optional Sampling
- Maximal Inequalities
- Crossing Lemmas
- Convergence
- Doob-Meyer Decomposition
- Weak Convergence
- Weak convergence and the Prohorov metric
- The topology of measure space
- Construction of Wiener Measure

**Additional References:**- D. Stroock, Probability Theory: an Analytic View, Cambridge University Press, 1993
- H. Royden, Real Analysis, McMillan, 1968

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

Department of Mathematics

University of Illinois
at Urbana-Champaign

1409 W Green St.

Urbana, IL 61801

r-sowers@math.uiuc.edu
(217) 333-6246