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**Bounds for the probability generating functional of a Gibbs point process.**
*(English)*
Zbl 1295.60061

Recently, in the papers [A. Baddeley and G. Nair, “Approximating the moments of a spatial point process”, Stat 1, No. 1, 18–30 (2012)] and [A. Baddeley and G. Nair, Electron. J. Stat. 6, 1155–1169 (2012; Zbl 1268.60063)], an approximation method that is fast to compute and accurate as verified by Monte Carlo methods have been proposed. There are, however, no theoretical results in this respect and hence no guarantees for accuracy in the most concrete models nor a quantification of the approximation error. The aim of the present paper is to derive rigorous lower and upper bounds for correlation functions and related quantities. The main result of the paper is the following statement.

{ Theorem 1.} Let \(\Xi\) be a stationary locally stable Gibbs process with intensity \(\lambda\) and local stability constant \(c^*\), and let \(g: {\mathbb R}^d\to [0, 1]\) be a function for which \(1-g\) has bounded support. Then \(1-\lambda G\leq {\operatorname E}\left(\prod_{y\in \Xi} g(y)\right) \leq 1 - {{\lambda}\over{c^*}} (1 - \exp{(-c^*\, G)})\), where \(G=\int_{{\mathbb R}^d} (1 - g(x))\, dx\), and \({\operatorname E}\) stands for expectation.

The idea for proving Theorem 1 is to replace the Gibbs process \(\Xi\) by a suitable Poisson process and to bound the error using Stein’s method. The rest of the paper is organized as follows. In Section 2, the authors introduce some notation and state the main result. In Section 3, bounds on the intensity are obtained. In Section 4, bounds on other summary statistics are derived. Section 5 contains the proof of Theorem 1.

{ Theorem 1.} Let \(\Xi\) be a stationary locally stable Gibbs process with intensity \(\lambda\) and local stability constant \(c^*\), and let \(g: {\mathbb R}^d\to [0, 1]\) be a function for which \(1-g\) has bounded support. Then \(1-\lambda G\leq {\operatorname E}\left(\prod_{y\in \Xi} g(y)\right) \leq 1 - {{\lambda}\over{c^*}} (1 - \exp{(-c^*\, G)})\), where \(G=\int_{{\mathbb R}^d} (1 - g(x))\, dx\), and \({\operatorname E}\) stands for expectation.

The idea for proving Theorem 1 is to replace the Gibbs process \(\Xi\) by a suitable Poisson process and to bound the error using Stein’s method. The rest of the paper is organized as follows. In Section 2, the authors introduce some notation and state the main result. In Section 3, bounds on the intensity are obtained. In Section 4, bounds on other summary statistics are derived. Section 5 contains the proof of Theorem 1.

Reviewer: Viktor Ohanyan (Erevan)

### MSC:

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

62M30 | Inference from spatial processes |

### Citations:

Zbl 1268.60063### Software:

spatstat
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\textit{K. Stucki} and \textit{D. Schuhmacher}, Adv. Appl. Probab. 46, No. 1, 21--34 (2014; Zbl 1295.60061)

### References:

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