a point process is characterized by its void probabilities - MathOverflow

a point process is characterized by its void probabilities

2010-09-21T12:54:19Z

Consider a planar point process $X$ and call $N_A = \text{Card}\big( X \cap A\big)$ the number of points inside the subset $A \subset \mathbb{R}^2$. If one knows the law of $(N_{A_1}, \ldots, N_{A_r})$ for any sets $A_1, \ldots, A_r$, then the process is completely characterized. I recently learned that it in fact suffices to know $f(A)=P(N_A=0)$ (called the void-probability function) for any set $A$ in order to completely characterize the law of $X$.

Intuitively, I do not understand why such a result is true. Indeed, the knowledge of the function $f$ brings some information in the correlation structure of the process $X$: nevertheless, I still fail to understand how the function $f$ can encode the whole correlation structure of the process. Any thoughts on this ?

Answer by Omer for a point process is characterized by its void probabilities

2010-09-21T15:59:59Z

This is only true for simple point processes (no duplicate points).

By the inclusion-exclusion principle, $f$ determines the joint distribution of several (disjoint) sets being empty or occupied. If the process is simple this allows recovering the law of $(N_{A_1},\dots,N_{A_r})$ as a limit over finer partitions.

Answer by yogesh for a point process is characterized by its void probabilities

2010-12-08T16:16:23Z

To add to Omer's concise explanation, the general result is known as Choquet's capacity theorem. It says that the void probabilities characterise any random closed set. Simple point processes are an example of random closed sets.