## The Annals of Applied Probability

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

Lothar Heinrich

#### Abstract

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.

#### Article information

Source
Ann. Appl. Probab., Volume 15, Number 1A (2005), 392-420.

Dates
First available in Project Euclid: 28 January 2005

https://projecteuclid.org/euclid.aoap/1106922332

Digital Object Identifier
doi:10.1214/105051604000001007

Mathematical Reviews number (MathSciNet)
MR2115047

Zentralblatt MATH identifier
1067.60002

#### Citation

Heinrich, Lothar. Large deviations of the empirical volume fraction for stationary Poisson grain models. Ann. Appl. Probab. 15 (2005), no. 1A, 392--420. doi:10.1214/105051604000001007. https://projecteuclid.org/euclid.aoap/1106922332

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