Syllabus for Math 451

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.

Provisional Schedule

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.

Grading Policy

Grades will be determined on the basis of homework, a midterm, and a final. The relative weights are:
Final:						40% of grade
Midterm:					30% of grade
Homework (4 assignments):			30% of grade
Total:						100% of grade
The midterm and final exam might be a take-home tests. If there is a regular final exam, it will be on Wednesday, May 8, 1996, from 1:30--4:30 PM.

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Richard Sowers (Home Page)
Department of Mathematics
University of Illinois at Urbana-Champaign
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Urbana, IL 61801
(217) 333-6246