**Instructor:**Richard Sowers**Office:**107 Altgeld 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:**Mondays, Wednesdays, and Fridays 1:30-2:30 P.M. and by appointment**Class meets:**Mondays, Wednesdays, and Fridays 3-3:50 P.M. in 1 Illini Hall.

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

This is the first-year graduate course in probability theory.
The goal of this course is to understand the basic tools of modern probability theory, and also to point out (in some small way)
some connections with other areas of mathematics and science (analysis, number theory, queuing
theory, and physics). Essentially, we shall look at probability theory as consisting of several
common ways of *modelling* randomness (i.e., Markov and martingale properties, independence, etc.)
and as a collection of *limit theorems* which allow one to extract some simple deterministic
properties from a more complicated random system (i.e., the laws of large numbers, central limit
theorems, ergodic theorems, martingale convergence theorems).
See also
the
Course Catalog Entry for Math 451 or the list of UIUC
Probability Theory Classes.

- Lecture 1: Introduction and probabilistic framework
- Lecture 2: Random variables
- Lecture 3: Expectation
- Lecture 4: Independence
- Lecture 5: Conditional probabilities and expectations
- Lecture 6: Convergence of random variables
- Lecture 7: Borel-Cantelli lemmas
- Lecture 8: Laws of large numbers
- Lecture 9: Large deviations
- Lecture 10: Central limit theorem
- Lecture 11: Martingales
- Lecture 12: Brownian motion
- Appendix: Kolmogorov's Extension Theorem

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

In comparison with previous years, we shall put less emphasis on some technicalities of the Central Limit Theorem. This will free up some time for discussion of large deviations and tightness.

Final: 40% of grade Midterm: 30% of grade Homework (4 assignments): 30% of grade Total: 100% of gradeThe midterm and final exam might be a take-home tests. If there is a regular final exam, it will be on

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Department of Mathematics

University of Illinois at Urbana-Champaign

1409 W Green St.

Urbana, IL 61801

r-sowers@math.uiuc.edu

(217) 333-6246