## 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.

### MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 62M30 Inference from spatial processes

Zbl 1268.60063

spatstat
Full Text:

### References:

 [1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Nat. Bureau Standards Appl. Math. Ser. 55). US Government Printing Office, Washington, DC. · Zbl 0171.38503 [2] Baddeley, A. and Nair, G. (2012). Approximating the moments of a spatial point process. Stat 1 , 18-30. [3] Baddeley, A. and Nair, G. (2012). Fast approximation of the intensity of Gibbs point processes. Electron. J. Statist. 6 , 1155-1169. · Zbl 1268.60063 [4] Baddeley, A. and Turner, R. (2005). Spatstat: an R package for analyzing spatial point patterns. J. Statist. Software 12 , 42pp. [5] Barbour, A. D. (1988). Stein’s method and Poisson process convergence. In A Celebration of Applied Probability (J. Appl. Prob. Spec. Vol. 25A ), Applied Probability Trust, Sheffield, pp. 175-184. · Zbl 0661.60034 [6] Barbour, A. D. and Brown, T. C. (1992). Stein’s method and point process approximation. Stoch. Process. Appl. 43 , 9-31. · Zbl 0765.60043 [7] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, General Theory and Structure , 2nd edn. Springer, New York. · Zbl 1159.60003 [8] Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32 , 844-865. · Zbl 1123.60309 [9] Mase, S. (1990). Mean characteristics of Gibbsian point processes. Ann. Inst. Statist. Math. 42 , 203-220. · Zbl 0723.60118 [10] Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes (Monogr. Statist. Appl. Prob. 100 ). Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1044.62101 [11] Møller, J. and Waagepetersen, R. P. (2007). Modern statistics for spatial point processes. Scand. J. Statist. 34 , 643-684. · Zbl 1157.62067 [12] Nguyen, X.-X. and Zessin, H. (1979). Integral and differential characterizations of the Gibbs process. Math. Nachr. 88 , 105-115. · Zbl 0444.60040 [13] Ruelle, D. (1969). Statistical Mechanics: Rigorous Results . W. A. Benjamin. New York. · Zbl 0177.57301 [14] Schuhmacher, D. and Stucki, K. (2013). Gibbs point process approximation: total variation bound using Stein’s method. To appear in Ann. Prob. Preprint available at http://uk.arxiv.org/abs/1207.3096. · Zbl 1322.60060 [15] Stoyan, D. (2012). Personal communication. [16] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications , 2nd edn. John Wiley, Chichester. · Zbl 0838.60002 [17] Xia, A. (2005). Stein’s method and Poisson process approximation. In An Introduction to Stein’s Method , Singapore University Press, pp. 115-181.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.