# a point process is characterized by its void probabilities

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 ?

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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.
Very nice. I'd known only one really nice example of inclusion-exclusion that could be presented in an introductory probability course with calculus as a prerequisite: the probability that a random permutation is a derangement approaches $1/e$ as the number of things being permuted grows. – Michael Hardy Dec 8 '10 at 16:25