Stein’s method for approximating complex distributions, With a view towards point processes

Abstract

We give an introduction to Stein’s method, a powerful technique for computing explicit error bounds for distributional approximation. The classical case of normal approximation is provided for initial motivation. Then the main part of this chapter is devoted to presenting the key concepts of Stein’s method in a much more general framework, where the approximating distribution Q and the space S it lives on can be almost arbitrary. This is particularly appealing for distributional approximation in stochastic geometry and spatial statistics. Rather than providing many concrete results, the emphasis of this chapter lies on conveying the techniques for developing Stein’s method on new state spaces S and for new approximating distributions. These techniques are elaborated in detail for the case where S is a space of point patterns and Q is the distribution of a Poisson process or a more general Gibbs process. Questions on how to measure distances between probability distributions on complicated spaces are also addressed. It is convenient if S is equipped with a suitable metric. We present several ideas and examples about performing statistical analyses on metric spaces.