a point process is characterized by its void probabilities - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:21:21Z http://mathoverflow.net/feeds/question/39491 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39491/a-point-process-is-characterized-by-its-void-probabilities a point process is characterized by its void probabilities Alekk 2010-09-21T12:54:19Z 2010-12-08T16:16:23Z <p>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 <code>$(N_{A_1}, \ldots, N_{A_r})$</code> 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 $X$.</p> <p>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 ?</p> http://mathoverflow.net/questions/39491/a-point-process-is-characterized-by-its-void-probabilities/39517#39517 Answer by Omer for a point process is characterized by its void probabilities Omer 2010-09-21T15:59:59Z 2010-09-21T15:59:59Z <p>This is only true for <b>simple</b> point processes (no duplicate points).</p> <p>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.</p> http://mathoverflow.net/questions/39491/a-point-process-is-characterized-by-its-void-probabilities/48653#48653 Answer by yogesh for a point process is characterized by its void probabilities yogesh 2010-12-08T16:16:23Z 2010-12-08T16:16:23Z <p>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. </p>