Some local approximation properties of simple point processes (Q957722)

A simple point process on a Borel space \(\mathbf S\) is a locally finite collection of distinct random elements \(\tau_k\) in \(\mathbf S\). A simple point process can be identified with the random counting measure \(\xi=\sum_k\delta_{\tau_k}\), assigning a unit mass \(\delta_{\tau_k}\) to each point \(\tau_k\). The aim of the present paper is to study a variety of asymptotic properties involving conditional distributions, given that \(\xi\) hits a small set \(B\subset \mathbf S\). In particular, the author is interested in the asymptotic behavior of the hitting probabilities \(P(\xi(B)>0)\) and conditional distributions \(P(\xi\in \cdot| \xi(B)>0)\), as \(B\) approaches a single point \(s\in \mathbf S\). The paper is organized as follows. A necessary technical background is provided by Section 2, where the author reviews the required definitions of Palm and Campbell measures and discusses the abstract notions of convergence and differentiation. Sections 3 and 4 form the core of the paper, where the author proves the basic approximation theorems.The results of Sections 3 and 4 are interesting, because of their analogy with properties of regenerative sets [see \textit{O. Kallenberg}, Probab. Theory Relat. Fields 125, No.~1, 1--41 (2003; Zbl 1022.60043) and exchangeable interval partitions in Stochastic Processes Appl. 94, No.~2, 241--270 (2001; Zbl 1053.60086)]. The last two sections are devoted to the mean convergence version of \(P(\xi(B)=1)\sim P(\xi(B)>0)\sim E\xi(B)\) (\(\sim\) denotes asymptotic equality as \(B\) shrinks to a single point) and some ratio limit theorems. In particular, Section 5 considers a case of a simple point process \(\xi\) on \(n\)-dimensional Euclidean space \(\mathbb R^n\).