2005/02/01 by Lothar Heinrich
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Random Matrices and Applications #math.PR
paper · pdf · doi:10.1214/105051604000001007
published as Annals of Applied Probability 2005, Vol. 15, No. 1A, 392-420 · Published at http://dx.doi.org/10.1214/105051604000001007 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2005/02/01 · arxiv created 2005/03/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function Ln(z)=|Wn|−1logEexpz|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 Eexpa|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.