**Instructor:**Richard Sowers**Office:**227B 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**Office Hours:**TBA**Class meets:**Mondays, Wednesdays, and Fridays 3-3:50 P.M. in 343 Altgeld Hall.

**Text:** R. Durrett Probability:
Theory and Examples , 2nd ed., 1995, Wadsworth & Brooks/Cole.

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). Our gestalt is that the goal of
probability is to understand *ASYMPTOTICS*; i.e., to extract
quantitative information from complicated probabilistic objects.
The first part of the course will be
dedicated to an understanding of what probability is and some
basic limit theorems. Then we will understand the topological
structure of probability measures (on Polish spaces). Then we will
make a rigorous study of sigma-fields, which correspond to
information. Finally, we will be able to discuss limit theorems.
We will concentrate on three such limit theorems; the law of large
numbers, the central limit theorem, and martingales (as the correct
way to understand the evolution of information).

- 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

You will be asked to turn in the above problems and (in a random manner) do them in class. Half of your grade will be based upon this, and half will be based upon the final exam, which will be take-home.

Department of Mathematics

University of Illinois at Urbana-Champaign

1409 W Green St.

Urbana, IL 61801

r-sowers@math.uiuc.edu

(217) 333-6246