We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function Ln(z)=|Wn|−1logEexp{z|Ξ∩Wn|} of the empirical volume fraction |Ξ∩Wn|/|Wn|, where |⋅| denotes the d-dimensional Lebesgue measure. Here Ξ=⋃i≥1(Ξi+Xi) denotes a d-dimensional Poisson grain model (also known as a Boolean model) defined by a stationary Poisson process Πλ=∑i≥1δXi with intensity λ>0 and a sequence of independent copies Ξ1,Ξ2,… of a random compact set Ξ0. For an increasing family of compact convex sets {Wn, n≥1} which expand unboundedly in all directions, we prove the existence and analyticity of the limit lim n→∞Ln(z) on some disk in the complex plane whenever Eexp{a|Ξ0|}<∞ for some a>0. Moreover, closely connected with this result, we obtain exponential inequalities and the exact asymptotics for the large deviation probabilities of the empirical volume fraction in the sense of Cramér and Chernoff.
Publié le : 2005-02-14
Classification:
Poisson grain model with compact grains,
volume fraction,
Cox process,
thermodynamic limit,
correlation measures,
cumulants,
large deviations,
Berry–Esseen bound,
Chernoff-type theorem,
60D05,
60F10,
60G55,
82B30
@article{1106922332,
author = {Heinrich, Lothar},
title = {Large deviations of the empirical volume fraction for stationary Poisson grain models},
journal = {Ann. Appl. Probab.},
volume = {15},
number = {1A},
year = {2005},
pages = { 392-420},
language = {en},
url = {http://dml.mathdoc.fr/item/1106922332}
}
Heinrich, Lothar. Large deviations of the empirical volume fraction for stationary Poisson grain models. Ann. Appl. Probab., Tome 15 (2005) no. 1A, pp. 392-420. http://gdmltest.u-ga.fr/item/1106922332/