## The Annals of Applied Probability

### Uniqueness of Unbounded Occupied and Vacant Components in Boolean Models

#### Abstract

We consider Boolean models in $d$-dimensional Euclidean space. Each point of a stationary, ergodic point process is the center of a ball with random radius. In this way, the space is partitioned into an occupied and a vacant region. We are interested in the number of unbounded occupied or vacant components that can coexist. We show that under very general conditions on the distribution of the radius random variable, there can be at most one unbounded component of each type. In case the point process is Poisson, we obtain uniqueness of the unbounded components without imposing any condition at all. Although we do not prove the necessity of the conditions to prove uniqueness, we obtain examples of stationary, ergodic point processes where the unbounded components are not unique when the conditions are violated. Finally, we discuss more general random shapes than just balls which are centered at the points of the point process.

#### Article information

Source
Ann. Appl. Probab., Volume 4, Number 3 (1994), 933-951.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177004978

Digital Object Identifier
doi:10.1214/aoap/1177004978

Mathematical Reviews number (MathSciNet)
MR1284992

Zentralblatt MATH identifier
0812.60093

JSTOR