# Syllabus for Math 451

Text: Personal Notes (ps)

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). 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.

## Provisional Schedule

1. Introduction and probabilistic framework
• Connections with measure theory
• Convergence of random variables
• a.s.
• convergence in L-p
• convergence in probability
• convergence in distribution
2. Information, Independence, and Conditioning
• Sigma-algebra manipulation
• The calculus of sigma-algebras
• Stopping times
• Independence
• Conditioning
3. Asymptotics: limit theorems
• The weak law of large numbers for square-integrable random variables
• Large deviations
• Central Limit Theorem
4. Martingales
• Optional Sampling
• Maximal Inequalities
• Crossing Lemmas
• Convergence
• Doob-Meyer Decomposition
5. Weak Convergence
• Weak convergence and the Prohorov metric
• The topology of measure space
6. Construction of Wiener Measure
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.