Insertion and deletion tolerance of point processes.(English)Zbl 1291.60101

Let $$\Pi$$ be a simple point process in $${\mathbb R}^d$$. Let $$\prec$$ denote the absolute continuity in law, i.e., for random variables $$X$$ and $$Y$$ taking values in the same measurable space, $$X\prec Y$$ if and only if $$\operatorname P(Y\in A)=0$$ implies $$\operatorname P(X\in A)=0$$ for all measurable $$A$$. Let $${\mathcal B}^d$$ denote the Borel $$\sigma$$-algebra in $${\mathbb R}^d$$ and $$L$$ be the Lebesgue measure. $$\Pi$$ is insertion-tolerant if for every $$B\in {\mathcal B}^d$$ with $$L(B)\in (0, \infty)$$ holds: if $$U$$ is uniformly distributed in $$B$$ and independent of $$\Pi$$, then $$\Pi + \delta_U\prec \Pi$$, where $$\delta_x$$ denotes the point measure at $$x\in {\mathbb R}^d$$. Let $${\mathbf M}$$ denote the space of simple point measures in $${\mathbb R}^d$$. The support of a measure $$\mu\in {\mathbf M}$$ is denoted by $$[\mu] =\{y\in {\mathbb R}^d:\,\, \mu(\{y\})=1\}$$. A $$\Pi$$-point is an $${\mathbb R}^d$$-valued variable $$Z$$ such that $$Z\in [\Pi]$$ a.s. A finite subprocess of $$\Pi$$ is a point process $${\mathcal F}$$ such that $${\mathcal F}({\mathbb R}^d)< \infty$$ and $$[{\mathcal F}] \subset [\Pi]$$ a.s. The point process $$\Pi$$ is called deletion-tolerant if for any $$\Pi$$-point $$Z$$ we have $$\Pi - \delta_Z\prec \Pi$$. The authors prove several equivalent formulations of insertion-tolerance and deletion-tolerance conditions. A point process is translation-invariant if it is invariant in law under the group of all translations of $${\mathbb R}^d$$.
Theorem 1. A translation-invariant point process $$\Pi$$ in $${\mathbb R}^d$$ is insertion-tolerant if and only if there exists $$B\in {\mathcal B}^d$$ with $$L(B)\in (0, \infty)$$ such that holds: if $$U$$ is uniformly distributed in $$B$$ and independent of $$\Pi$$, then $$\Pi + \delta_U\prec \Pi$$.
The Boolean continuum percolation model for point processes is defined as follows( see [R. Meester and R. Roy, Continuum percolation. Cambridge: Cambridge Univ. Press (1996; Zbl 0858.60092)]): For $$R>0$$ and $$\mu\in {\mathbf M}$$, consider the set $$O(\mu)= \cup_{x\in[\mu]} B(x, R)$$, where $$B(x, R)$$ is the open ball of radius $$R$$ with center $$x$$. $$O(\mu)$$ is called the occupied region. The connected components of $$O(\mu)$$ are called clusters.
Theorem 2 (continuum percolation). Let $$\Pi$$ be a translation-invariant ergodic insertion-tolerant point process in $${\mathbb R}^d$$. For any $$R>0$$, the occupied region $$O(\Pi)$$ has at most one unbounded cluster a.s.
The proof of Theorem 2 is similar to the uniqueness proofs in Chapter 7.

MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60D05 Geometric probability and stochastic geometry 82B43 Percolation

Zbl 0858.60092
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