On the typical point

The concept of the typical point is important in point process theory. In the wireless context, it is often concretized to the typical user, typical receiver, typical vehicle, etc. The typical point is an abstraction in the sense that it is not a point that is selected in any deterministic fashion from the point process, neither is it an arbitrary point. Also, speaking of “a typical point” is misleading, since it suggests that there exist a number of such typical points in the process (or perhaps even in a realization thereof), and we can choose one of them to “obtain a typical point”. This does not work – there is nothing “typical” about a point selected in a deterministic fashion, let alone an arbitrary point.
That said, if the total number of points is finite, we can, loosely speaking, argue that the typical point is obtained by choosing one of the points uniformly at random. Even this is not trivial since picking a point in a single realization of the point process does generally not produce the desired result, for example of the point process is not ergodic. Many realizations need to be considered, but then the question arises how exactly to pick a point uniformly from many realizations. If the number of points is infinite, any attempt of picking a point uniformly at random is bound to fail because there is no way to select a point uniformly from infinitely many, in much the same way that we cannot select an integer uniformly at random.

So how do we arrive at the typical point? If the point process is stationary and ergodic, we can generate the statistics of the typical point by averaging those of each individual point in a single realization, observed in an increasingly large observation window. This leads to the interpretation of the typical point as a kind of “average point”. Similarly, we may find that the “typical American male” weighs 89.76593 kg, which does not imply that there exists an individual with that exact weight but means that if we weighed every(male)body or a large representative sample, we would obtain that average weight. For general (stationary) point processes, we obtain the typical point by conditioning on a point to exist at a certain location, usually the origin o. Upon averaging over the point process (while holding on to this point at the origin), that point becomes the typical point. The distribution of this conditioned point process is the Palm distribution. In the case of the Poisson process, conditioning on a point at o is the same as adding a point at o. This equivalence is called Slivnyak’s theorem.

In the non-stationary case, the typical point at location x may have different statistical properties than the typical point at another location y. As a result, there exist Palm measures (or distributions) for each location in the support of the point process. In this case, the formal definition of the Palm measure is given by the Radon-Nikodym derivative (with respect to the intensity measure) of the Campbell measure. In the Poisson case, Slivnyak’s theorem still applies.

3 thoughts on “On the typical point

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