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Consider two independent point processes in the unit square $[0,1]^2$. The two point processes are identically independent and typically binomial/Poisson. One, say $\Phi^*$, is used to generate a Voronoi tessellation of $N^*$ cells. The other one, $\Phi$ with $N$ points, is used to build the statistics of the counts of points of $\Phi$ in the Voronoi cells associated with $\Phi^*,N^*$.

Gilbert's argument (Ann. Math. Stat., 1962) for a randomly picked given point in plane to be contained in a Voronoi cell of area $s$ is that, known the p.d.f. of cell areas -- be it $f(s)$ --, then the probability of $f(s|X=1)=sE[s]^{-1}f(s)$ where $X$ is a random variable that expresses the count of points in $s$. How can I generalize the argument to any $X$ value? In principle, if I consider $k$ points randomly drawn and independently, I thought of considering $f^k(s|X=1)$, but it does not match simulations. Any suggestion?

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  • $\begingroup$ Can you give more details about the reference you quote? Voronoi cells, in the simplest case, are associated to a set of points in the plane (or more generally a metric space). They need not have equal areas. Please give more details. en.wikipedia.org/wiki/Voronoi_diagram $\endgroup$ – Liviu Nicolaescu May 22 at 10:53
  • $\begingroup$ @LiviuNicolaescu The reference is to the classic Gilbert's paper in Ann. Math. Stat. 1962 (in particular the statement is at p. 963, bottom, second paragraph of Section 4). $\endgroup$ – maurizio May 22 at 10:56
  • $\begingroup$ @LiviuNicolaescu I should mention, that in my case I suppose to know $f(s)$ -- the cell area distribution of the random Voronoi tesselation -- the analytical form of this p.d.f. is not known, but can be well approximated by Gamma distributions. $\endgroup$ – maurizio May 22 at 11:02
  • $\begingroup$ Thanks! I will check it out $\endgroup$ – Liviu Nicolaescu May 22 at 18:32
  • $\begingroup$ Could you explain the model a bit more? By definition of the Voronoi cells, there is one and only one point in each cell. I'm assuming their seeds are given by a Poisson point process? Are you sampling more points once the domains are chosen? And if so, how? $\endgroup$ – Pierre PC May 29 at 4:51

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