Statistics 930: Probability Theory
Professor J. Michael Steele
Prerequisites. This course is designed for graduate students in Statistics, OPIM,
mathematics, applied mathematics, and related fields. (It is also open to advanced
undergraduates who have had a course in real analysis that included Lebesgue
integration.)
Course Objectives. The course is designed to provide a professional and rigorous
understanding of the most fundamental theorems of probability theory. The central
focus will be on sums of independent random variables, but the course will also
deal to a lesser extent with martingales, stationary processes (Ergodic theory), and
Markov processes. The level and extent of the course is well captured by the text:
Probability:Theory and Examples (Fourth Edition) by Rick Durrett.
Special Emphasis on Problem Solving. The course will give much attention
to problem solving. The intention is to engage probability theory as a living subject
with discoveries just waiting to be made.
Course Topics:
Probability Spaces, Integration, and Independence: Sigma fields and
why they matter. Tools of Expectation (Monotone Convergence Theorem,
Dominated Convergence Theorem, Fatou’s Lemma), Modes of convergence,
especially convergence in probability and convergence with probability one,
Inequalities of Markov, Chebyshev, and others, the Borel Cantelli Lemmas,
Lovasz Local Lemma
Theory of the Law of Large Numbers: Easy SLLNs, Etemadi’s Proof of
SLLN, Kolmogorov’s Three Series Theorem.
Characteristic Functions: Fundamental facts, including Fourier inversion.
Applications in distribution theory. Essen’s inequality.
Central Limit Theory: Proof of the CLT and its refinements using charac-
teristic functions. Proof of the CLT by Lindeberg’s method, Proof of the
CLT by Stein’s method.
Martingale Methods: Basic martingale theory including Doob’s maximal
inequality and several proofs of the martingale convergence theorem. The
CLT for martingales.
Stationary Processes: The Ergodic theorem and several proofs. CLT for
stationary processes.
Text, Policies, Etc.
Texts: Probability:Theory and Examples (Fourth Edition) by Rick Durrett
Homework: Regular homework will be assigned and graded. Homework so-
lutions will not be provided; one is encouraged to ask about solutions,
ideas, hints, etc. in class or office hours.
Grading: Grades are based on the homework (30%), midterm (20%) and a
final exam (50%).
Auditors: Auditors are welcome. Homework by auditors will not be graded.
More Information?
Office Hours: Please see Steele’s personal web page for current contact in-
formation.
Still More?: Google “Steele Probability Course” etc. for the Course Blog.
All course information will be posed on the blog.