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|.1logEexp{z|..Wn|} of the empirical volume fraction |..Wn|/|Wn|, where |.| denotes the d-dimensional Lebesgue measure. Here .=.i.1(.i+Xi) denotes a d-dimensional Poisson grain model (also known as a Boolean model) defined by a stationary Poisson process ..=.i.1.Xi with intensity .>0 and a sequence of independent copies .1,.2,. of a random compact set .0. 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 Eexp{a|.0|}<. 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.