ST553 Half Unit
Probability and Mathematical Statistics II
This information is for the 2022/23 session.
Teacher responsible
Prof Konstantinos Kardaras
Availability
This course is available on the MPhil/PhD in Statistics. This course is available with permission as an outside option to students on other programmes where regulations permit.
Pre-requisites
Probability and Mathematical Statistics I is a pre-requisite.
Course content
This course provides instruction in advanced topics in probability and mathematical statistics, mainly based on martingale theory. It is a continuation of Probability and Mathematical Statistics I. The following topics will in particular be covered:
- Conditional expectation revisited; linear regression; martingales and first examples.
- Concentration inequalities; dimension reduction; log-Sobolev inequalities.
- Martingale transforms; optional sampling theorem; convergence theorems.
- Sequential testing; backwards martingales; law of large numbers; de Finetti’s theorem.
- Markov chains; recurrence; reversibility; foundations of MCMC.
- Ergodic theory.
- Brownian motion; quadratic variation; stochastic integration.
- Stochastic differential equations; diffusions; filtering.
- Bayesian updating; Ergodic diffusions; Langevin samplers.
- Brownian bridge; empirical processes; Kolmogorov-Smirnov statistic.
Teaching
This course will be delivered through a combination of classes, lectures and Q&A sessions totalling a minimum of 30 hours across Lent Term. This course includes a reading week in Week 6 of Lent Term.
Formative coursework
Students will be expected to produce 9 problem sets in the LT.
Weekly problem sets that are discussed in subsequent seminars. The coursework that will be used for summative assessment will be chosen from a subset of these problems.
Indicative reading
- Williams, D. (1991). Probability with Martingales. Cambridge University Press.
- Durrett, R. (2019). Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics.
- Karatzas, I, Shreve S. (1991). Brownian motion and Stochastic Calculus. Springer GTM.
- Shao, J. (2007). Mathematical Statistics. Springer Texts in Statistics.
- Keener, R. (2010). Theoretical Statistics. Springer Texts in Statistics.
Assessment
Exam (70%, duration: 3 hours, reading time: 10 minutes) in the summer exam period.
Coursework (30%).
Three of the homework problem sets will be submitted and marked as assessed coursework.
Key facts
Department: Statistics
Total students 2021/22: 2
Average class size 2021/22: 2
Value: Half Unit
Course selection videos
Some departments have produced short videos to introduce their courses. Please refer to the course selection videos index page for further information.
Personal development skills
- Problem solving
- Application of numeracy skills
- Specialist skills