Large Deviations of the Empirical Volume Fraction for Stationary Poisson Grain Models

We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function Ln(z) = |Wn| log E exp{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 Boolean model) defined by a stationary Poisson process Πλ = ∑ i≥1 δXi 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 limn→∞ Ln(z) on some disk in the complex plane whenever E exp{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.