This course aims to provide an introduction on Markov chains in discrete time.

The main content includes:
  • Definition and basic properties of Markov chains, transition probabilities and Chapman-Kolmogorov equations;
  • Classification of states: Communicating and absorbing states; Recurrence and transience;
  • Hitting probabilities and mean hitting times; survival probability for birth and death chains;
  • Random walks;
  • Stationary distribution and ergodic theorem;
  • Markov Chain Monte Carlo methods; Gibbs sampler.

PrerequisiteSolid knowledge in fundamental probability theory. "Stochastics I" (or equivalent course) is recommended.

The course will be held in English.

If you are interested to attend the course, please ask for the enrolment key at your earliest convenience via email to: nguyen[at]math.uni-sb.de.

The focus is on studying the convergence behavior of local averaging methods (e.g. kernel estimates and nearest neighbor estimates) and of empirical risk minimization. These methods are widely used for solving regression problems and in pattern recognition.

Topics:

  • Introduction to the regression problem and to pattern recognition
  • Local averaging methods (e.g., kernel smoothing, k-nearest neighbor)
  • Concentration inequalities (Hoeffding, Bernstein)
  • Sample splitting
  • Empirical risk minimization
  • Vapnik-Chervonenkis inequality
  • Combinatorial aspects of the Vapnik-Chervonenkis theory
  • Neural networks

Prerequisites: Good knowledge in probability theory as taught e.g. in the course "Stochastics I".

This course provides an introduction to the theory of stochastic processes. Topics include:

  • Conditioning on sigma-fields
  • Stochastic processes: basic notions and examples
  • Construction of stochastic processes via the Kolmogorov extension theorem
  • Martingales
  • Martingal convergence theorems and applications
  • Optional sampling theorems
  • Brownian motion
  • Regularity of Brownian paths and the law of the iterated logarithm
  • Skorokhod embedding
  • Donsker's invariance principle
  • (Strong) Markov property
  • Applications of Brownian motion to partial differential equations

Prerequisites are Analysis I-III, Linear Algebra I-II, Stochastics I (or equivalent courses).

The aim of this course is to provide an introduction to continuous-time Markov processes. The included topics are:

  • Definition and existence of Markov processes
  • Strong Markov property
  • Lévy processes as examples of (strong) Markov processes
  • Contraction semigroup and Infinitesimal generator
  • Feller semigroup, Feller processes and the existence of their cadlag modifications.
  • Martingale problems

The prerequisite course is "Stochastics I". 

The course "Stochastics II" is helpful but not obliged.