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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

Citations:

Zbl 0858.60092
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