13 Stationarity
So far, we have defined general point processes and their first- and second-order structure.
However, these quantities can be very complex in full generality, as they depend on locations \(u\) and pairs \((u,v)\).
To make progress, we often impose structural assumptions.
One of the most important of these is stationarity.
13.1 Definition
A point process \(X\) on \(\mathbb{R}^d\) is said to be stationary if its distribution is invariant under translations.
Formally, for any vector \(h \in \mathbb{R}^d\),
\[ X \overset{d}{=} X + h, \]
where
\[ X + h := \{u + h: u \in X\} \]
13.2 Intuition
Stationarity means:
The statistical properties of the process do not depend on where you are in space.
If you “shift the entire pattern”, nothing changes probabilistically.
13.3 Consequences for first order structure
Recall the intensity function:
\[ \lambda(u) = \lim_{|du|\to 0} \frac{\mathbb{E}[N(du)]}{|du|}. \]
Under stationarity:
\[ \lambda(u) = \lambda \ \text{constant} \]
That is, stationarity implies constant intensity.
It is important to note the reverse is NOT true in general.
Example…
13.4 Consequences for Second-Order Structure
Recall the second-order product density:
\[ \lambda^{(2)}(u,v). \]
Under stationarity, this can only depend on the relative displacement between \(u\) and \(v\):
\[ \lambda^{(2)}(u,v) = \lambda^{(2)}(u - v). \]
That is, the dependence structure is translation invariant.
13.5 Pair correlation function
Recall the pair correlation function:
\[ g(u,v) = \frac{\lambda^{(2)}(u,v)}{\lambda(u)\lambda(v)}. \]
Under stationarity:
- \(\lambda(u) = \lambda\)
- \(\lambda^{(2)}(u,v) = \lambda^{(2)}(u-v)\)
So:
\[ g(u,v) = g(u - v). \]
Thus, the PCF depends only on the vector difference between points.
13.6 Why This Matters
Stationarity simplifies everything:
| Quantity | General case | Stationary case |
|---|---|---|
| Intensity | \(\lambda(u)\) | \(\lambda\) |
| Second-order | \(\lambda^{(2)}(u,v)\) | \(\lambda^{(2)}(u-v)\) |
| PCF | \(g(u,v)\) | \(g(u-v)\) |
This reduction is crucial:
- Instead of functions of two variables \((u,v)\)
- We now work with functions of a single displacement vector
13.7 Limitations
Stationarity is often unrealistic in practice:
- Environmental heterogeneity
- Spatial trends (e.g. population density, elevation)
- Large-scale gradients
In such cases, we instead work with:
- Inhomogeneous models
- Residual analysis after detrending
13.8 Summary
Stationarity is the assumption that:
- The process looks the same everywhere in space
- The intensity is constant
- Dependence depends only on relative displacement
This assumption dramatically simplifies second-order analysis, and forms the foundation for further simplifications such as isotropy.