Say we have a homogeneous PPP with rate $\lambda$ in the 2-D plane $\mathbb R^2$. In one realization of the PPP we get the points $\phi=\{x_1,x_2,...,x_i,...\}$.
Now we generate the Voronoi cells with the $k$ nearest points ($k$ order Voronoi cell WiKi, Demo).
Then consider the number of these cells in which a given point $x_i\in \phi$ takes part. Can we claim that the expected number of cells in which a point takes part is the same for any point $\{x_1,x_2,...,x_i,...\}$?
It seems intuitive and I tried with crude simulation and it seems to hold. But I don't know how to define expectation because the points will change with each trial. Then also how do we take expectation of a area the PPP only defines probabilities of points.
I think this problem might be Related to Correlations between neighboring Voronoi cells.
Many thanks for any suggestions or a reference.
The probability measure to take an expectation over is the "Palm Measure".
Basically, you will need to define a notion of a "typical" point x_0 say of the PPP. Then to define the expectation you are asking, crudely you will have to check the number of cells in which this typical point lies in each realization and take an average.
The notion of a "typical" point is made rigorous through Palm Measures and Palm calculus.
However, since the underlying process is Poisson, the computation is easy due to Slivnyak's theorem. The typical point in the case of homogenous PPP is the origin, i.e compute the expected number of cells the origin is a part of given that the origin is part of the PPP. This computation is easy by applying Slivnyak's theorem i.e $\mathbf{P} { \{ \phi \in A | origin \in \phi \}} = \mathbf{P} \{ \phi - \delta_0 \in A \}$ where $\phi$ is the PPP and A is the desired property (an event in the sigma field).