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:

1. Conditional expectation revisited; linear regression; martingales and first examples.
2. Concentration inequalities; dimension reduction; log-Sobolev inequalities.
3. Martingale transforms; optional sampling theorem; convergence theorems.
4.  Sequential testing; backwards martingales; law of large numbers; de Finetti’s theorem.
5. Markov chains; recurrence; reversibility; foundations of MCMC.
6. Ergodic theory.
7. Brownian motion; quadratic variation; stochastic integration.
8. Stochastic differential equations; diffusions; filtering.
9. Bayesian updating; Ergodic diffusions; Langevin samplers.
10. Brownian bridge; empirical processes; Kolmogorov-Smirnov statistic.

10 hours of seminars and 20 hours of lectures in the Winter Term.

This course has a reading week in Week 6 of Winter Term.

Students will be expected to produce 9 problem sets in the WT.

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.

1. Williams, D. (1991). Probability with Martingales. Cambridge University Press.
2. Durrett, R. (2019). Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics.
3. Karatzas, I, Shreve S. (1991). Brownian motion and Stochastic Calculus. Springer GTM.
4. Shao, J. (2007). Mathematical Statistics. Springer Texts in Statistics.
5. Keener, R. (2010). Theoretical Statistics. Springer Texts in Statistics.

Exam (70%), duration: 180 Minutes, reading time: 10 minutes in the Spring exam period

Problem sets (30%)

Three of the homework problem sets will be submitted and marked as assessed coursework.