Probability Theory: The Coupling Method

Coupling is a powerful method in probability theory through which random variables can be compared with each other. Coupling has been applied in a broad variety of contexts, e.g. to prove limit theorems, to derive inequalities, or to obtain approximations. The present course is intended for master students and PhD students. A basic knowledge of probability theory is required, as well as some familiarity with measure theory. The course first explains what coupling is and what general framework it fits into. After that a number of applications are described. These applications illustrate the power of coupling and at the same time serve as a guided tour through some key areas of modern probability theory. Examples include: random walks, card shuffling, Poisson approximation, Markov chains, correlation inequalities, percolation, interacting particle systems, and diffusions.