Process level moderate deviations for stabilizing functionals

Peter Eichelsbacher; Tomasz Schreiber

ESAIM: Probability and Statistics (2010)

  • Volume: 14, page 1-15
  • ISSN: 1292-8100

Abstract

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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. 


How to cite

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Eichelsbacher, Peter, and Schreiber, Tomasz. "Process level moderate deviations for stabilizing functionals." ESAIM: Probability and Statistics 14 (2010): 1-15. <http://eudml.org/doc/250824>.

@article{Eichelsbacher2010,
abstract = {
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. 
},
author = {Eichelsbacher, Peter, Schreiber, Tomasz},
journal = {ESAIM: Probability and Statistics},
keywords = {Moderate deviations; random Euclidean graphs; random sequential packing.; moderate deviation; random Euclidean graph; random sequential packing},
language = {eng},
month = {2},
pages = {1-15},
publisher = {EDP Sciences},
title = {Process level moderate deviations for stabilizing functionals},
url = {http://eudml.org/doc/250824},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Eichelsbacher, Peter
AU - Schreiber, Tomasz
TI - Process level moderate deviations for stabilizing functionals
JO - ESAIM: Probability and Statistics
DA - 2010/2//
PB - EDP Sciences
VL - 14
SP - 1
EP - 15
AB - 
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. 

LA - eng
KW - Moderate deviations; random Euclidean graphs; random sequential packing.; moderate deviation; random Euclidean graph; random sequential packing
UR - http://eudml.org/doc/250824
ER -

References

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  1. Y. Baryshnikov and J.E. Yukich, Gaussian limits for random measures in geometric probability. Ann. Appl. Probab.15 (2005) 213–253.  
  2. Y. Baryshnikov, P. Eichelsbacher, T. Schreiber and J.E. Yukich, Moderate Deviations for some Point Measures in Geometric Probability. Ann. Inst. H. Poincaré44 (2008) 422–446; electronically available on the arXiv, math.PR/0603022.  
  3. F. Comets, Grandes déviations pour des champs de Gibbs sur ℤ d (French) [ Large deviation results for Gibbs random fields on ℤ d ] . C. R. Acad. Sci. Paris Sér. I Math.303 (1986) 511–513.  
  4. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Second edition. Springer (1998).  
  5. H. Föllmer and S. Orey, Large Deviations for the Empirical Field of a Gibbs Measure. Ann. Probab.16 (1988) 961–977.  
  6. H.-O. Georgii, Large Deviations and Maximum Entropy Principle for Interacting Random Fields on ℤ d . Ann. Probab.21 (1993) 1845–1875.  
  7. H.-O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction. Probab. Theory Relat. Fields99 (1994) 171–195.  
  8. H.-O. Georgii and H. Zessin, Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Relat. Fields96 (1993) 177–204.  
  9. P. Hall, Introduction to the Theory of Coverage Processes. Wiley, New York (1988).  
  10. I.S. Molchanov, Limit Theorems for Unions of Random Closed Sets. Lect. Notes Math. 1561. Springer (1993)  
  11. S. Olla, Large Deviations for Gibbs Random Fields. Probab. Theor. Rel. Fields77 (1988) 343–357.  
  12. M.D. Penrose, Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab.33 (2005) 1945–1991.  
  13. M.D. Penrose, Gaussian Limits for Random Geometric Measures, Electron. J. Probab.12 (2007) 989–1035.  
  14. M.D. Penrose and J.E. Yukich, Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab.11 (2001) 1005–1041.  
  15. M.D. Penrose and J.E. Yukich, Limit theory for random sequential packing and deposition. Ann. Appl. Probab.12 (2002) 272–301.  
  16. M.D. Penrose and J.E. Yukich, Weak laws of large numbers in geometric probability. Ann. Appl. Probab.13 (2003) 277–303.  
  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.  
  18. D. Stoyan, W. Kendall and J. Mecke, Stochastic Geometry and Its Applications. Second edition. John Wiley and Sons (1995).  

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