Count of random points belonging to a Voronoi cell in the unit square

Question

I am considering a Voronoi tessellation in the unit square $V \in [0,1]^2$ for $G$ points uniformly randomly distributed. Then I am considering $N$ other points also randomly distributed uniformly in $V$ (and independently of $G$), and I want to find an expression for the p.d.f. of the counts of these points in the Voronoi cells, as I change $G$ and $N$.

This is how I am reasoning. Because $G\gg 1$ in my case, I can approximate the distribution of Voronoi cell areas by the Gamma distribution $\mathrm{Gamma}(x;\nu;\nu^{-1})$ [Kiang, 1966], where $x$ is the cell area $s$ normalized w.r.t. the mean cell area $\rho^{-1}$, i.e. $x=\rho s$. Moreover, following [Pineda et al., 2006], I get an excellent fit for my data when $\nu=3.575$.

Now the harder part. In the classical paper by Gilbert [1962,p. 963], it is reasoned that the probability of a randomly-picked point in the plane to belong to a cell of area $s$ is $p^*(s)=\rho s \cdot p(s)$ where $p(s)$ is the p.d.f. of cell areas, i.e. $\mathrm{Gamma}(\rho s; \nu, \nu^{-1})$ in my case. Once again, it is convenient to work in terms of the normalized cell area $x$, whereby I can define the p.d.f. that given one point, it belongs to a cell of size $x$, by $f(x|n=1)=x\cdot \mathrm{Gamma}(x;\nu,\nu^{-1})$.

If everything worked out fine, the probability that $k$ randomly-picked points belong to the same cell of size $x$ would then be $f^k(x|n=1)$ (because of the independent drawing). Moreover, assuming that $f(k|x)$ - the probability that given a cell of size $x$, it contains $k$ points randomly-drawn from $N$ - is a $\mathrm{Binomial}(k,N,x)$, it would follow that $f_k(k) = f(k|x) p(x)f^{-k}(x|n=1)$.

This reasoning is apparently seriously wrong. But I cannot figure out where I am wrong. If then everything makes sense up to this point, what am I missing/doing that I cannot correctly fit my data?

Answer 1

Ok, I did not find an analytical formula for the above, except for an integral expression, obtained by marginalizing $f(k,x)$, since there is not a direct way to compute $f(x|n=k)$, that is the p.d.f. that a cell of size $x$ contains at least $k$ points. So in my case, $f_k(k) = \int_0^G f(k|x)f(x)dx$ provides a correct estimation for the distribution of random points in a Voronoi cell in a random tessellation of $G$ cells in unit square.

Note: $f(x|k)$ is not trivial to estimate, because it certainly relates to the probability that $k$ independent points are each found in a cell of size $x$, i.e. $f^k(x|1)$, but in addition, we also need to estimate the probability that such cell is the same for all $k$ points, or in other words, the probability that randomly picking $k$ points, the all fall in the same cell.