Process level moderate deviations for stabilizing functionals
ESAIM: Probability and Statistics, Volume 14 (2010), p. 1-15

Functionals of spatial point process often satisfy a weak spatial dependence condition known as stabilization. In this paper we prove process level moderate deviation principles (MDP) for such functionals, which is a level-3 result for empirical point fields as well as a level-2 result for empirical point measures. The level-3 rate function coincides with the so-called specific information. We show that the general result can be applied to prove MDPs for various particular functionals, including random sequential packing, birth-growth models, germ-grain models and nearest neighbor graphs.

DOI : https://doi.org/10.1051/ps:2008027
Classification:  60F05,  60D05
Keywords: moderate deviations, random euclidean graphs, random sequential packing
@article{PS_2010__14__1_0,
     author = {Eichelsbacher, Peter and Schreiber, Tomasz},
     title = {Process level moderate deviations for stabilizing functionals},
     journal = {ESAIM: Probability and Statistics},
     publisher = {EDP-Sciences},
     volume = {14},
     year = {2010},
     pages = {1-15},
     doi = {10.1051/ps:2008027},
     zbl = {pre05872984},
     mrnumber = {2640365},
     language = {en},
     url = {http://www.numdam.org/item/PS_2010__14__1_0}
}
Eichelsbacher, Peter; Schreiber, Tomasz. Process level moderate deviations for stabilizing functionals. ESAIM: Probability and Statistics, Volume 14 (2010) pp. 1-15. doi : 10.1051/ps:2008027. http://www.numdam.org/item/PS_2010__14__1_0/

[1] Y. Baryshnikov and J.E. Yukich, Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005) 213-253. | Zbl 1068.60028

[2] Y. Baryshnikov, P. Eichelsbacher, T. Schreiber and J.E. Yukich, Moderate Deviations for some Point Measures in Geometric Probability. Ann. Inst. H. Poincaré 44 (2008) 422-446; electronically available on the arXiv, math.PR/0603022. | Numdam | Zbl 1175.60015

[3] F. Comets, Grandes déviations pour des champs de Gibbs sur d (French) [ Large deviation results for Gibbs random fields on d ] . C. R. Acad. Sci. Paris Sér. I Math. 303 (1986) 511-513. | Zbl 0606.60035

[4] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Second edition. Springer (1998). | Zbl 1177.60035

[5] H. Föllmer and S. Orey, Large Deviations for the Empirical Field of a Gibbs Measure. Ann. Probab. 16 (1988) 961-977. | Zbl 0648.60028

[6] H.-O. Georgii, Large Deviations and Maximum Entropy Principle for Interacting Random Fields on d . Ann. Probab. 21 (1993) 1845-1875. | Zbl 0790.60031

[7] H.-O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction. Probab. Theory Relat. Fields 99 (1994) 171-195. | Zbl 0803.60097

[8] H.-O. Georgii and H. Zessin, Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Relat. Fields 96 (1993) 177-204. | Zbl 0792.60024

[9] P. Hall, Introduction to the Theory of Coverage Processes. Wiley, New York (1988). | Zbl 0659.60024

[10] I.S. Molchanov, Limit Theorems for Unions of Random Closed Sets. Lect. Notes Math. 1561. Springer (1993) | Zbl 0790.60015

[11] S. Olla, Large Deviations for Gibbs Random Fields. Probab. Theor. Rel. Fields 77 (1988) 343-357. | Zbl 0621.60031

[12] M.D. Penrose, Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab. 33 (2005) 1945-1991. | Zbl 1087.60022

[13] M.D. Penrose, Gaussian Limits for Random Geometric Measures, Electron. J. Probab. 12 (2007) 989-1035. | Zbl 1153.60015

[14] M.D. Penrose and J.E. Yukich, Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001) 1005-1041. | Zbl 1044.60016

[15] M.D. Penrose and J.E. Yukich, Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002) 272-301. | Zbl 1018.60023

[16] M.D. Penrose and J.E. Yukich, Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003) 277-303. | Zbl 1029.60008

[17] A. Rényi, Théorie des éléments saillants d'une suite d'observations, in Colloquium on Combinatorial Methods in Probability Theory. Mathematical Institut, Aarhus Universitet, Denmark (1962), pp. 104-115. | Zbl 0139.35303

[18] D. Stoyan, W. Kendall and J. Mecke, Stochastic Geometry and Its Applications. Second edition. John Wiley and Sons (1995). | Zbl 1155.60001