Stein's method

Abstract:

Distributional limit theorems such as the Central Limit Theorem are important in statistics and probabilistic combinatorics. These kinds of theorems are usually stated asymptotically, with a topological notion of distributional convergence called weak convergence. A shortcoming of these theorems is that they are not quantitative. That is, they give no understanding of the rate of convergence.

Stein's method is a technique for bounding the distance between distributions in a variety of metrics consistent with weak convergence. We will introduce the idea with a simple proof of a quantitative version of the central limit theorem. We then turn to a more involved problem in probabilistic combinatorics. To approach this problem we will use Stein's method in combination with the idea of an exchangeable pair.