Large deviations of the empirical volume fraction for stationary Poisson grain models
| dc.creator | Heinrich, Lothar | |
| dc.date | 2005-03-23 | |
| dc.date.accessioned | 2026-07-07T05:18:15Z | |
| dc.date.available | 2026-07-07T05:18:15Z | |
| dc.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. | |
| dc.description | 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) | |
| dc.identifier | https://arxiv.org/abs/math/0503479 | |
| dc.identifier | http://arxiv.org/abs/math/0503479 | |
| dc.identifier | Annals of Applied Probability 2005, Vol. 15, No. 1A, 392-420 | |
| dc.identifier | doi:10.1214/105051604000001007 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/74597 | |
| dc.subject | Probability | |
| dc.subject | 60D05\sep60F10 (Primary) 60G55\sep82B30 (Secondary) | |
| dc.title | Large deviations of the empirical volume fraction for stationary Poisson grain models | |
| dc.type | text |