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 L_n(z)=|W_n|^{-1}\logE\exp{z|\Xi\cap W_n|} of the empirical volume fraction |\Xi\cap W_n|/|W_n|, where |\cdot| denotes the d-dimensional Lebesgue measure. Here \Xi=\bigcup_{i\ge1}(\Xi_i+X_i) denotes a d-dimensional Poisson grain model (also known as a Boolean model) defined by a stationary Poisson process \Pi_{\lambda}=\sum_{i\ge1}\delta_{X_i} with intensity \lambda >0 and a sequence of independent copies \Xi_1,\Xi_2,... of a random compact set \Xi_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|\Xi_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 Cram\'er and Chernoff. 
Lothar Heinrich