Large deviations of the empirical volume fraction for stationary Poisson grain models
Abstract
Description
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.
Published at http://dx.doi.org/10.1214/105051604000001007 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Published at http://dx.doi.org/10.1214/105051604000001007 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)