The Palm Distributions P₀ and P₀^!

 In later sections of this paper we need Palm distributions of a different kind than that given in (3.2). Roughly speaking, such a (global) Palm distribution is the conditional distribution of (X,M) under the condition that there are points of the point process at a given set of locations. We begin with the simplest case conditioning on the event that (X,M) has a point at a fixed location, say at the origin.

Let A be a set of realizations of (X,M) and put  

$$P_0(A) = \left( \lambda \nu(B) \right)^{-1} \, \Exp \, \sum... ...X_k \in B} \, \bfind \left( \{ (X_n - X_k, M_n)\} \in A \right)$$ (4)

where B is an arbitrary fixed Borel set with $0 < \nu(B) < \infty$.By $\{(X_n - X_k, M_n),n=0,\pm 1, \ldots\}$ we denote the shifted point process which we obtain after shifting all points of (X,M) by X_k. In particular, the point X_k of (X,M) is shifted into the origin. Clearly, the sum in (3.4) counts the shifted realizations lying in A. Since we assume that (X,M) is stationary, the value P₀(A) defined in (3.4) does not depend on the choice of B. Moreover, in many cases of practical interest we have  

$$P_0(A) = \lim_{\varepsilon \downarrow 0} \, \Prob \left( (X,M) \in A \, \vert \, X(b(0,\varepsilon)) = 1 \right)$$ (5)

which motivates the interpretation of P₀ as conditional distribution of (X,M) under the condition that at the origin there is a point of the point process.

If (X,M) is ergodic and the sequence $\{B_n\}$ is an averaging sequence, then analogous to (3.3)  

$$P_0(A) = \lim_{i \to \infty} \, \frac{\sum_{k:X_k \in B_i} \... ...ind \left( \{ (X_n-X_k,M_n)\} \in A \right)}{X_M(B_i \times J)}$$ (6)

which gives the motivation to say that P₀(A) is the probability of the event A, seen from the typical point of (X,M).

We remark that the notion of the Palm mark distribution defined in (3.2) can be embedded into the more general concept given by (3.4). Namely, if $A = \{ \mbox{the mark of the point at the origin belongs to }I\}$,then

 

P₀(A) = P_M(I)

because in this case $P_0(A) = (\lambda \nu(B))^{-1} \Exp \, \sum_{k\,:\, X_k \in B} \, \bfind(M_k\in I) \, = \, \frac{\lambda_I}{\lambda} \, = \, P_M(I).$

It is clear that a marked point process with distribution P₀ is not stationary since, under P₀, there is a point at the origin with probability 1. However, if (X,M) is stationary and isotropic, then a marked point process with distribution P₀ is isotropic as well.

A further important notion is that of the reduced Palm distribution P₀^! which is given by  

$$P_0^!(A) = (\lambda \nu(B))^{-1}\Exp \, \sum_{k\,:\, X_k \in B} \, \bfind \left( \{ (X_n-X_k,M_n), n\neq k \} \in A \right).$$ (8)

This means that under P₀^!, the point at the origin is not counted.

Clearly, by omitting the marks in (3.4) and (3.8), one can define the distributions P₀ and P₀^! of an (unmarked) stationary point process $X=\{X_n\}$ in the same way. A famous result of the theory of stationary point processes is the following theorem of Slivnyak: X is a Poisson process if and only if

 

P₀^! = P

where P denotes the (unconditional) distribution of X. In other words, the Palm distribution P₀ of a homogeneous Poisson process is obtained simply by adding a point at the origin. See e.g. Møller (1994) for a proof of it.

Further details on Palm distributions of stationary marked point processes on the plane and in higher-dimensional spaces can be found e.g. in Daley and Vere-Jones (1988), König and Schmidt (1992), Møller (1994), Stoyan et al. (1995). We also remark that their initial application was in queueing theory; see e.g. Baccelli and Brémaud (1994), Brandt et al. (1990), Franken et al. (1982).