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

Abstract

We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function L_n(z)=|W_n|⁻¹logEexp{z|Ξ∩W_n|} of the empirical volume fraction |Ξ∩W_n|/|W_n|, where |⋅| denotes the d-dimensional Lebesgue measure. Here Ξ=⋃_i≥1(Ξ_i+X_i) denotes a d-dimensional Poisson grain model (also known as a Boolean model) defined by a stationary Poisson process Π_λ=∑_i≥1δ_Xi 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 {W_n, n≥1} which expand unboundedly in all directions, we prove the existence and analyticity of the limit lim _n→∞L_n(z) on some disk in the complex plane whenever Eexp{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.