The Palm Mark Distribution

 Suppose that (X,M) is stationary, with intensity $\lambda = \Exp X_M ([0,1]^2 \times J)$ such that $0 < \lambda < \infty$. Define for $B \subset \RL^2$, $I \subset J$ 

$$\lambda (B \times I) = \Exp X_M (B \times I)$$ (1)

which is called the intensity measure of (X,M). By the stationarity of (X,M) we have $\lambda (B \times I) = \lambda_I \nu(B)$,where $\lambda_I = \lambda ([0,1]^2 \times I)$ is the so-called I-intensity of (X,M), i.e., $\lambda_I$ is the expected number of points per unit area with marks in I. Now, the so-called Palm mark distribution P_M is defined by the ratio  

$$P_M(I) = \frac{\lambda_I}{\lambda}.$$ (2)