Open Access
February 2005 Large deviations of the empirical volume fraction for stationary Poisson grain models
Lothar Heinrich
Ann. Appl. Probab. 15(1A): 392-420 (February 2005). DOI: 10.1214/105051604000001007


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≥1i+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 Ξ12,… 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{a0|}<∞ 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.


Download Citation

Lothar Heinrich. "Large deviations of the empirical volume fraction for stationary Poisson grain models." Ann. Appl. Probab. 15 (1A) 392 - 420, February 2005.


Published: February 2005
First available in Project Euclid: 28 January 2005

zbMATH: 1067.60002
MathSciNet: MR2115047
Digital Object Identifier: 10.1214/105051604000001007

Primary: 60D05 , 60F10
Secondary: 60G55 , 82B30

Keywords: Berry–Esseen bound , Chernoff-type theorem , correlation measures , Cox process , Cumulants , large deviations , Poisson grain model with compact grains , thermodynamic limit , volume fraction

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.15 • No. 1A • February 2005
Back to Top