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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