Introduction

Conditioning and the typical point

The Palm probability or Palm measure in point process theory is the probability of an event given that the point process contains a point at some location. It also formalizes the notion of the “typical point” of the process. Informally, the typical point results from a selection procedure in which every point has the same chance of being selected. This idea needs to be made mathematically precise, especially in infinite point processes. For example, a point chosen according to some sampling procedure, such as the one closest to the origin, is not typical, because it has been selected in a specific, deterministic manner. Intuitively, the Palm distribution is the conditional point process distribution given that a point (the typical point) exists at a specific location.

This type of conditioning is sometimes referred to as interior conditioning, since the conditioning is on x ∈ Φ and the question is how the point process looks outside of x. In contrast, the Papangelou conditional intensity is based on exterior conditioning, since the conditioning is on ℝd {x}, and the question is how likely it is to have a point at x. The two concepts are dual to each other.

If the point process is atomic, as the die process, conditioning on having a point at the location of one of the atoms causes no dificulty.