## Gibbs point process approximation: total variation bounds using Stein’s method.(English)Zbl 1322.60060

Point processes (fields) on a compact metric space $$\mathcal X$$ are considered. Such a point process is determined by a probability measure on the set $$\mathcal N$$ of all finite subsets of this space. This probability measure depends on their family of conditional probabilities. A class of Gibbs processes is defined in terms of a partial ordering on the set $$\mathcal N$$. The best known Gibbs process is constructed on the base of pairwise interaction of points from its realizations. This interaction is determined by some non-negative function $$\varphi$$ on $$\mathcal X \times \mathcal X$$. The Gibbs process is said to be a hard core process if there exists some $$a > 0$$ such that $$\varphi (x_1 , x_2 ) = 1$$ when $$\text{dist}(x_1 , x_2 ) \leq a$$, and $$\varphi(x_1 , x_2 ) = 0$$ when $$\text{dist}(x_1 , x_2 ) > a$$. Another example of a Gibbs process is called Lennard-Jones process. It has an exponentially decreasing function $$\varphi$$ as a function of distance between its arguments. A distance on $$\mathcal N$$ and the corresponding distance on the set of measures on $$\mathcal N$$ are defined. The distance between two Gibbs distributions is proved to be estimated from above in terms of a distance between their conditional intensities. These estimates permit to determine an approximation of general type Gibbs processes with the help of more investigated classes (for example, hard core Gibbs processes, or Lennard-Jones processes). The obtained results are applied for the solution of the coupling problem for two processes of birth and death. The main method of investigation used in the paper is a method of Stein with the proofs are being reduced to the solution of some integral-differential equation.

### MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G60 Random fields 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text:

### References:

 [1] Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47 601-619. · Zbl 0848.60051 [2] Barbour, A. D. (1988). Stein’s method and Poisson process convergence. J. Appl. Probab. 25A 175-184. · Zbl 0661.60034 [3] Barbour, A. D. and Brown, T. C. (1992). Stein’s method and point process approximation. Stochastic Process. Appl. 43 9-31. · Zbl 0765.60043 [4] Barbour, A. D. and Chen, L. H. Y., eds. (2005). An Introduction to Stein’s Method . Singapore Univ. Press, Singapore. · Zbl 1072.62007 [5] Barbour, A. D. and Månsson, M. (2002). Compound Poisson process approximation. Ann. Probab. 30 1492-1537. · Zbl 1033.60059 [6] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. J. Roy. Statist. Soc. Ser. B 36 192-236. · Zbl 0327.60067 [7] Besag, J., Milne, R. and Zachary, S. (1982). Point process limits of lattice processes. J. Appl. Probab. 19 210-216. · Zbl 0488.60065 [8] Brown, T. C. and Greig, D. (1994). Correction to: “Stein’s method and point process approximation” [ Stochastic Process. Appl. 43 (1992), no. 1, 9-31; MR1190904 (93k:60120)] by A. D. Barbour and Brown. Stochastic Process. Appl. 54 291-296. · Zbl 0765.60043 [9] Chen, L. H. Y. and Xia, A. (2004). Stein’s method, Palm theory and Poisson process approximation. Ann. Probab. 32 2545-2569. · Zbl 1057.60051 [10] Dai Pra, P. and Posta, G. (2013). Entropy decay for interacting systems via the Bochner-Bakry-Émery approach. Electron. J. Probab. 18 no. 52, 21. · Zbl 1286.60097 [11] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II , 2nd ed. Springer, New York. · Zbl 1159.60003 [12] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes . Wiley, New York. · Zbl 0592.60049 [13] Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes , 3rd ed. Oxford Univ. Press, New York. · Zbl 1015.60002 [14] Kallenberg, O. (1986). Random Measures , 4th ed. Akademie-Verlag, Berlin. · Zbl 0345.60032 [15] Kallenberg, O. (2002). Foundations of Modern Probability : Probability and Its Applications , 2nd ed. Springer, New York. · Zbl 0996.60001 [16] Kondratiev, Y. and Lytvynov, E. (2005). Glauber dynamics of continuous particle systems. Ann. Inst. Henri Poincaré Probab. Stat. 41 685-702. · Zbl 1085.60074 [17] Kondratiev, Y., Pasurek, T. and Röckner, M. (2012). Gibbs measures of continuous systems: An analytic approach. Rev. Math. Phys. 24 1250026, 54. · Zbl 1267.82029 [18] Møller, J. (1989). On the rate of convergence of spatial birth-and-death processes. Ann. Inst. Statist. Math. 41 565-581. · Zbl 0702.60072 [19] Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes . Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1044.62101 [20] Preston, C. (1975). Spatial birth-and-death processes. In Proceedings of the 40 th Session of the International Statistical Institute ( Warsaw , 1975) 371-391. · Zbl 0379.60082 [21] Ruelle, D. (1969). Statistical Mechanics : Rigorous Results . W. A. Benjamin, New York. · Zbl 0177.57301 [22] Schuhmacher, D. (2009). Distance estimates for dependent thinnings of point processes with densities. Electron. J. Probab. 14 1080-1116. · Zbl 1196.60089 [23] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability ( Univ. California , Berkeley , CA , 1970 / 1971), Vol. II : Probability Theory 583-602. Univ. California Press, Berkeley, CA. · Zbl 0278.60026 [24] Stucki, K. and Schuhmacher, D. (2014). Bounds for the probability generating functional of a Gibbs point process. Adv. in Appl. Probab. 46 21-34. · Zbl 1295.60061 [25] Xia, A. and Zhang, F. (2012). On the asymptotics of locally dependent point processes. Stochastic Process. Appl. 122 3033-3065. · Zbl 1247.60069
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.