Stationary Marked Point Processes

  The important concept of the typical point of a stationary point process which has been mentioned in Section 2.1 in connection with the nearest neighbor distance distribution function D(r), will be repeatedly used in this paper. The aim of the present section is to make this concept mathematically precise. There are two possible ways of doing this, either by using the approach via the Palm mark distribution of a stationary marked point process, or by using a spatial ergodic theorem (assuming then that the underlying point process is stationary and ergodic).

The marked point process $(X,M) = \{(X_1,M_1), (X_2,M_2),\ldots\}$is said to be stationary if the translated marked point processes $\{(X_1-x,M_1),(X_2-x,M_2),\ldots\}$ have the same distribution for all $x \in \RL^2$. However, like in the case of an (unmarked) point process considered in Section 2.1, it is mathematically more convenient to formulate the notion of stationarity in terms of counting variables. Let $X_M (B \times I)$ denote the random number of points of the marked point process (X,M) which lie spatially in the set B and have their marks from the set I. Notice that I belongs to a certain family of subsets of the space J of all possible marks which can be quite general. For example, in connection with random tessellations of the plane considered in Section 5, this so-called mark space is chosen to be the family of all closed subsets of $\RL^2$. Now, (X,M) is stationary if the random vectors $(X_M((B_1+x)\times I_1), \ldots, X_M((B_n+x)\times I_n))$ have the same distribution for all $x \in \RL^2$ and for all finite sequences of bounded Borel sets $B_1,\ldots,B_n$ and of all admissible mark sets $I_1,\ldots,I_n$.

In addition to stationarity one frequently assumes ergodicity. This property ensures that one point pattern observed in a large window B is sufficient to obtain statistically secure results. A sufficient condition for ergodicity is the following mixing property: (X,M) is called mixing if

$$\Prob((X,M) \in A, \; (X-x,M) \in A^{\prime}) \to \Prob ((X,M) \in A) \Prob((X,M) \in A^{\prime})$$

as $\vert\vert x\vert\vert \to \infty$, where $A, A^{\prime}$ are sets of point-process realizations. Intuitively speaking, the above condition means that the marked point processes (X,M) and (X-x,M) become more and more independent as the length ||x|| of the shift vector x increases unboundedly. For the precise definition of ergodicity we refer to Daley and Vere-Jones (1988).