The Annals of Applied Probability

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

Lothar Heinrich

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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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1106922332

Digital Object Identifier
doi:10.1214/105051604000001007

Mathematical Reviews number (MathSciNet)
MR2115047

Zentralblatt MATH identifier
1067.60002

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F10: Large deviations
Secondary: 60G55: Point processes 82B30: Statistical thermodynamics [See also 80-XX]

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

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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