# Probability that a Voronoi cell contains exactly k random points

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?

• 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 – Liviu Nicolaescu May 22 at 10:53
• @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). – maurizio May 22 at 10:56
• @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. – maurizio May 22 at 11:02
• Thanks! I will check it out – Liviu Nicolaescu May 22 at 18:32
• 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? – Pierre PC May 29 at 4:51