Large deviations of the empirical volume fraction for stationary Poisson grain models

Annals of Applied Probability

Volumn 15, Issue 1 A, 2005, Pages 392-420

  Heinrich, Lothar ^a  

^a UNIVERSITY OF AUGSBURG   (Germany)

Author keywords

Berry Esseen bound; Chernoff type theorem; Correlation measures; Cox process; Cumulants; Large deviations; Poisson grain model with compact grains; Thermodynamic limit; Volume fraction

Indexed keywords

EID: 14544298828     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/105051604000001007     Document Type: Article

References (22)

1. DALEY, D. J. and VERE-JONES, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
2. DEMBO, A. and ZEITOUNI, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
3. FRITZ, J. (1970). Generalization of McMillan's theorem to random set functions. Studia Sci. Math. Hangar. 5 369-394.
4. GEORGII, H.-O. and ZESSIN, H. (1993). Large deviations and maximum entropy principle for marked point random fields. Probab. Theory Related Fields 96 177-204.
5. GÖTZE, F., HEINRICH, L. and HIPP, C. (1995). m-dependent random fields with analytic cumulant generating function. Scand. J. Statist. 22 183-195.
6. GREENBERG, W. (1971). Thermodynamic states of classical systems. Comm. Math. Phys. 22 259-268.
7. HALL, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.
8. HEINRICH, L. (1992). On existence and mixing properties of germ-grain models. Statistics 23 271-286.
9. HEINRICH, L. and MOLCHANOV, I. S. (1999). Central limit theorem for a class of random measures associated with germ-grain models. Adv. in Appl. Probab. 31 283-314.
10. HEINRICH, L. and SCHMIDT, V. (1985). Normal convergence of multidimensional shot noise and rates of this convergence. Adv. in Appl. Probab. 17 709-730.
11. IVANOFF, G. (1982). Central limit theorems for point processes. Stochastic Process. Appl. 12 171-186.
12. MASE, S. (1982). Asymptotic properties of stereological estimators of volume fraction for stationary random sets. J. Appl. Probab. 19 111-126.
13. MATHERON, G. (1975). Random Sets and Integral Geometry. Wiley, New York.
14. MATTHES, K., KERSTAN, J. and MECKE, J. (1978). Infinitely Divisible Point Processes. Wiley, Chichester.
15. MOLCHANOV, I. S. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester.
16. PIAU, D. (1999). Large deviations and Young measures for a Poissonian model of biphased material. Ann. Appl. Probab. 9 706-718.
17. ROTA, G.-C. (1964). On the foundations of combinatorial theory: I. Theory of Möbius functions. Z. Wahrsch. Verw. Gebiete 2 340-368.
18. RUELLE, D. (1964). Cluster properties of correlation functions of classical gases. Rev. Mod. Phys. 36 580-586.
19. RUELLE, D. (1969). Statistical Mechanics: Rigorous Results. Benjamin, Amsterdam.
20. SAULIS, L. and STATULEVIČIUS, V. (1991). Limit Theorems for Large Deviations. Kluwer Academic, Dordrecht.
21. STATULEVIČIUS, V. A. (1966). On large deviations. Z. Wahrsch. Verw. Gebiete 6 133-144.
22. STOYAN, D., KENDALL, W. S. and MECKE, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Wiley, Chichester.