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|^{-1}\logE\exp{z|Ξ\cap W_n|} of the empirical volume fraction |Ξ\cap W_n|/|W_n|, where |\cdot| denotes the d-dimensional Lebesgue measure. Here Ξ=\bigcup_{i\ge1}(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=\sum_{i\ge1}δ_{X_i} with intensity λ>0 and a sequence of independent copies Ξ_1,Ξ_2,... of a random compact set Ξ_0. For an increasing family of compact convex sets {W_n, n\ge1} which expand unboundedly in all directions, we prove the existence and analyticity of the limit lim_{n\to\infty}L_n(z) on some disk in the complex plane whenever E\exp{a|Ξ_0|}<\infty 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.