URL study guide
https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=2MMS30&collegejaar=2025&taal=enDescription
Random walks (combinatorial identities, recurrence and transience (Pólya’s theorem), Donsker’s Theorem); martingales (optional stopping theorem, martingale convergence theorems, inequalities)Objectives
Acquiring knowledge of two main types of stochastic processes which play a fundamental role in probability theory and its applications: Random walks and martingales. Students are expected both to understand the proofs of the treated results, and to be able to apply these results in applications.At the end of the course, the student can
- Calculate basic and more advanced probabilities about simple one-dimensional random walk through the application of path counting and time inversion techniques.
- Relate random walk hitting times to the final position of the walk, its maximum and the total progeny of a branching process.
- Determine the asymptotics of random walk probabilities through application of Donsker’s invariance principle and Stirling’s formula.
- Demonstrate an active understanding of qualitative properties of d-dimensional random walk such as recurrence/transience and basic limit theorems.
- Argue by means of a mathematical proof involving the standard rules of conditional expectation whether a given stochastic process is a sub-/super/martingale.
- Evaluate expectations of stopping times and hitting probabilities through a correct formal application of the optional stopping and Lebesgue convergence theorems.
- Identify an almost surely convergent sub-/super-/martingale by checking L^1-boundedness and examine when it is additionally convergent in L^1 by inference of the limit or proof of uniform integrability.
- Derive conventration inequalities for standard martingales with bounded differences through application of the Azuma-Hoeffding inequality.
- Apply Doob’s submartingale inequality to bound the probability that the sample path of a submartingale exceeds a given value, possibly involving previous transformation of the process.