I do not understand the difference between a stationary and a homogeneous point process. The definitions I found are as follows:
- A process is stationary if the entire configuration of the process is invariant under translation.
- A process is homogeneous if the mean number of points in each set A is given by $\alpha \cdot \parallel A \parallel$, where $\alpha \geq 0$ is a constant.
These definitions appear the same to me. Could anyone give me an example of a process that is stationary but not homogeneous or the other way around? Thank you!!
A stationary point process is homogeneous (if the mean number of points in the unit square is finite); see, for example, Proposition 8.2 in "Lectures on the Poisson Process" by Günter Last and Mathew Penrose (Cambridge University Press, 2017).
However, the converse does not hold in general.
Consider, for example, a square lattice and add to each point an independent random vector that is uniformly distributed on the unit square. The resulting point process (a perturbed lattice) is homogeneous with $\alpha = 1$, but it is not stationary. There is certainly a point within the unit square, but there maybe between 0 and 4 points in the unit square if the point process is translated by a vector (0.5,0.5).
In special cases, including Poisson point processes, a point process is stationary if and only if it is homogeneous (see Proposition 8.3 in Last, Penrose 2017).