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

top

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

top

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

top
  1. Y. Baryshnikov and J.E. Yukich, Gaussian limits for random measures in geometric probability. Ann. Appl. Probab.15 (2005) 213–253.  Zbl1068.60028
  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.  Zbl1175.60015
  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.  Zbl0606.60035
  4. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Second edition. Springer (1998).  Zbl0896.60013
  5. H. Föllmer and S. Orey, Large Deviations for the Empirical Field of a Gibbs Measure. Ann. Probab.16 (1988) 961–977.  Zbl0648.60028
  6. H.-O. Georgii, Large Deviations and Maximum Entropy Principle for Interacting Random Fields on d . Ann. Probab.21 (1993) 1845–1875.  Zbl0790.60031
  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.  Zbl0803.60097
  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.  Zbl0792.60024
  9. P. Hall, Introduction to the Theory of Coverage Processes. Wiley, New York (1988).  Zbl0659.60024
  10. I.S. Molchanov, Limit Theorems for Unions of Random Closed Sets. Lect. Notes Math. 1561. Springer (1993)  Zbl0790.60015
  11. S. Olla, Large Deviations for Gibbs Random Fields. Probab. Theor. Rel. Fields77 (1988) 343–357.  Zbl0621.60031
  12. M.D. Penrose, Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Probab.33 (2005) 1945–1991.  Zbl1087.60022
  13. M.D. Penrose, Gaussian Limits for Random Geometric Measures, Electron. J. Probab.12 (2007) 989–1035.  Zbl1153.60015
  14. M.D. Penrose and J.E. Yukich, Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab.11 (2001) 1005–1041.  Zbl1044.60016
  15. M.D. Penrose and J.E. Yukich, Limit theory for random sequential packing and deposition. Ann. Appl. Probab.12 (2002) 272–301.  Zbl1018.60023
  16. M.D. Penrose and J.E. Yukich, Weak laws of large numbers in geometric probability. Ann. Appl. Probab.13 (2003) 277–303.  Zbl1029.60008
  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.  
  18. D. Stoyan, W. Kendall and J. Mecke, Stochastic Geometry and Its Applications. Second edition. John Wiley and Sons (1995).  Zbl0838.60002

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.