A Spatial Ergodic Theorem

 Consider a so-called averaging sequence $B_1,B_2, \ldots \, \subset \RL^2$ of test sets such that B_n is convex and compact, $B_n \subset B_{n+1}$for each $n \geq 1$, and

$$\sup \{ r \geq 0\, : \, b(x,r) \subset B_n \mbox{ for some }x \} \to \infty\qquad n \to \infty,$$

where b(x,r) denotes the circle with radius r and center at x. Then in many cases of practical interest we have with probability 1  

$$P_M(I) = \lim_{n \to \infty} \, \frac{X_M(B_n \times I)}{X_M(B_n \times J)}.$$ (3)

We remark that a sufficient condition for (3.3) is that (X,M) is ergodic. In particular, (3.3) holds if (X,M) is an independently marked Poisson process. Other examples where (3.3) is true are marked point processes which are induced by a homogeneous Poisson process (or an arbitrary stationary ergodic point process) via a shift-compatible mapping of its realizations; see Daley and Vere-Jones (1988). Such marked point processes will be considered e.g. in Section 5, in connection with random tessellations of the plane.

Equation (3.3) gives the motivation to say that P_M is the mark distribution of the typical point of (X,M). For keeping the terminology simple, we use this way of speaking also in the case when we do not assume that the limit in (3.3) is constant with probability 1. Note, however, that this interpretation of the Palm mark distribution is then not fully justified.

In the case when the marks are nonnegative random variables, the expectation of P_M will be denoted by $\overline{m} = \int_0^{\infty} \, m \; P_M(dm)$ and called the expected mark of the typical point.