Point-Map-Probabilities of a Point Process and Mecke's Invariant Measure Equation

A compatible point-shift $F$ maps, in a translation invariant way, each point of a stationary point process $\Phi$ to some point of $\Phi$. It is fully determined by its associated point-map, $f$, which gives the image of the origin by $F$. It was proved by J. Mecke that if $F$ is bijective, then the Palm probability of $\Phi$ is left invariant by the translation of $-f$. The initial question motivating this paper is the following generalization of this invariance result: in the non-bijective case, what probability measures on the set of counting measures are left invariant by the translation of $-f$? The point-map probabilities of $\Phi$ are defined from the action of the semigroup of point-map translations on the space of Palm probabilities, and more precisely from the compactification of the orbits of this semigroup action. If the point-map probability exists, is uniquely defined, and if it satisfies certain continuity properties, it then provides a solution to this invariant measure problem. Point-map probabilities are objects of independent interest. They are shown to be a strict generalization of Palm probabilities: when $F$ is bijective, the point-map probability of $\Phi$ boils down to the Palm probability of $\Phi$. When it is not bijective, there exist cases where the point-map probability of $\Phi$ is singular with respect to its Palm probability. A tightness based criterion for the existence of the point-map probabilities of a stationary point process is given. An interpretation of the point-map probability as the conditional law of the point process given that the origin has $F$-pre-images of all orders is also provided. The results are illustrated by a few examples.Comment: 35 pages, 2 figure

TVS-cone metric spaces as a special case of metric spaces

There have been a number of generalizations of fixed point results to the so called TVS-cone metric spaces, based on a distance function that takes values in some cone with nonempty interior (solid cone) in some topological vector space. In this paper we prove that the TVS-cone metric space can be equipped with a family of mutually equivalent (usual) metrics such that the convergence (resp. property of being Cauchy sequence, contractivity condition) in TVS sense is equivalent to convergence (resp. property of being Cauchy sequence, contractivity condition) in all of these metrics. As a consequence, we prove that if a topological vector space $E$ and a solid cone $P$ are given, then the category of TVS-cone metric spaces is a proper subcategory of metric spaces with a family of mutually equivalent metrics (Corollary 3.9). Hence, generalization of a result from metric spaces to TVS-cone metric spaces is meaningless. This, also, leads to a formal deriving of fixed point results from metric spaces to TVS-cone metric spaces and makes some earlier results vague. We also give a new common fixed point result in (usual) metric spaces context, and show that it can be reformulated to TVS-cone metric spaces context very easy, despite of the fact that formal (syntactic) generalization is impossible. Apart of main results, we prove that the existence of a solid cone ensures that the initial topology is Hausdorff, as well as it admits a plenty of convex open sets. In fact such topology is stronger then some norm topology.Comment: 14 page