## Annals of Probability

- Ann. Probab.
- Volume 45, Number 3 (2017), 1723-1751.

### Point-map-probabilities of a point process and Mecke’s invariant measure equation

François Baccelli and Mir-Omid Haji-Mirsadeghi

#### Abstract

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 nonbijective 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.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 3 (2017), 1723-1751.

**Dates**

Received: August 2014

Revised: January 2016

First available in Project Euclid: 15 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1494835229

**Digital Object Identifier**

doi:10.1214/16-AOP1099

**Mathematical Reviews number (MathSciNet)**

MR3650413

**Zentralblatt MATH identifier**

1375.60091

**Subjects**

Primary: 60G10: Stationary processes 60G55: Point processes 60G57: Random measures

Secondary: 60G30: Continuity and singularity of induced measures 60F17: Functional limit theorems; invariance principles

**Keywords**

Point process stationarity palm probability point-shift point-map allocation rule vague topology mass transport principle dynamical system $\omega$-limit set

#### Citation

Baccelli, François; Haji-Mirsadeghi, Mir-Omid. Point-map-probabilities of a point process and Mecke’s invariant measure equation. Ann. Probab. 45 (2017), no. 3, 1723--1751. doi:10.1214/16-AOP1099. https://projecteuclid.org/euclid.aop/1494835229