Probability I (Stat GR6301) - Fall 2016
This course is for PhD students in Statistics only.
- Instructor: Marcel Nutz
- Office
hours: Tuesdays by appointment; Thursdays 11:25-12:15, Room
910 SSW
- Meeting time: Tuesdays and Thursdays 8:40-9:55
- Room: Engineering 253
- Teaching Assistant: Florian Stebegg
- TA office hours:
Thursdays 10:00-11:00, Room 906 SSW
General Information
- Texts: No text is required. Some recommended
texts:
- J. Jacod and P. Protter, Probability
Essentials, 2nd edition,
2004.
- R. Durrett, Probability: Theory and Examples, 4th
edition, 2010.
- P. Billingsley, Probability
and Measure, 3rd edition, 1995.
- D. Williams, Probability with
Martingales, 1990.
- Grading: The homework and midterm will each count for
20% of the final grade, and the final exam will count for the remaining
60%.
- Midterm Examination: There will be one midterm, on Thursday, October 13th,
during class time. Students will be excused from the midterm only
because of serious illness or another emergency of similar gravity; a
note from a doctor or from a Dean will be required.
- Final Examination: The
final exam takes place on Tuesday,
December 20th, 9:00-12:00 in 903 SSW.
- Conflicts:
If you have a conflict with any of the exams (for example, due to a
religious holiday), please contact the instructor as soon as possible.
- Disabilities: Students who may need disability-related
accommodations should contact the instructor as soon as possible. Also,
stop by the Office of Disability Services to register for support
services.
Exercises
There will be weekly homework assignments,
generally assigned on Tuesdays and due the following Tuesday
by 11:30 a.m. in Florian's mailbox (10th floor). Your lowest
two homework
scores
will be dropped, and it is expected that this will take care
of any
homework you may fail to hand in due to illness, travel, bad luck, and
so on. You are encouraged to discuss the homework with fellow students,
but
you must write your solutions individually. Exercises and Solutions are
posted on Courseworks.
Syllabus
The syllabus is tentative and not exhaustive.
- Measure, independence, integral, basic inequalities, convergence
theorems, L^p spaces, uniform integrability
- Conditional expectation, projection in L^2
- Change-of-variable formula, Borel-Cantelli, Dynkin systems and
monotone classes
- Kernels, Fubini, Ionescu-Tulcea, disintegration
- Completion, Caratheodory extensions and Lebesgue-Stieltjes
measures
- Convergence in distribution, Skorokhod's theorem, Helly's
theorem, Prokhorov's theorem
- Strong and weak laws of large numbers
- Central limit theorem, Lindeberg's theorem
- Cramer-Wold device, multivariate Gaussian distribution
- Characteristic functions, method of moments
- Lebesgue decomposition and Radon-Nikodym theorem, Hahn and Jordan
decomposition of a signed measure
- Random series: Kolmogorov and Levy inequalities, Levy's theorem,
Three Series Theorem
- Large deviations: Chernov bound and Cramer's theorem