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) log E exp{z\Xi boolean AND Wn\} of the empirical volume fraction \Xi boolean AND W-n\/\W-n\, where \ . \ denotes the d-dimensional Lebesgue measure. Here Xi = U-igreater than or equal to1 (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) = Sigma(igreater than or equal to1) delta(Xi) 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 greater than or equal to 1} which expand unboundedly in all directions, we prove the existence and analyticity of the limit lim(n-->infinity) L-n(z) on some disk in the complex plane whenever E exp{a\Xi(0)\} < infinity 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 Cramer and Chernoff.