# Stationarity assumption for a point process

as I understand stationarity of a point process implies that it is invariant under translation. Imagine one has numerous realizations of the same point process. Due to the nature of this process, it can be observed only in a bounded region (i.e. islands, cell), the are can differ from observation to observation and so the number of points (i.e. it is random variable). Would it be correct to envision this process as stationary (since an infinite number of realization of this process is possible and nothing is in stone, but the concept of boundary) or it is just a finite process? Would it be true that non-homogenous process (clustering) is non-stationary? I am a bit confused and would appreciate hearing your opinion. Thanks!

Stationarity implies that the statistical parameters of an underlying process do not vary over space (or time), and is often formally described as "invariance under translation" (not to be confused with "invariance under rotation": isotropy). Homogeneity has a more specific meaning that pertains to the assumption that the average "intensity" $(\lambda)$ of a point process (i.e, the average number of points per unit area) is constant. In other words, it is akin to the concept of first order stationarity in the context of a point process.
For a homogeneous Poisson point process $(\mathbf{X})$, the expected number of points falling in some sub-region $B$ is $\mathbb{E} [n(\mathbf{X}{\bigcap}B]) =\lambda\cdot|B|$. The observed counts in different regions may vary because they represent random samples, but each sample is drawn from the same underlying distribution (with the same mean intensity)...in other words, a point process that is first-order stationary.