Berry–Esseen bounds and Cramér-type large deviations for the volume distribution of Poisson cylinder processes

Abstract

A stationary Poisson cylinder process (d;k) cyl is composed of a stationary Poisson process ofk-flats in R d that are dilated by i.i.d. random compact cylinder bases taken from the corresponding orthogonal complement. We study the accuracy of normal approximation of thed-volumeV (d;k) % of the union set of (d;k) cyl that covers%W as the scaling factor % becomes large. Here W is some fixed compact star-shaped set containing the origin as an inner point. We give lower and upper bounds of the variance of V (d;k) % that exhibit long-range dependence within the union set of cylinders. Our main results are sharp estimates of the higher-order cumulants of V (d;k) % under the assumption that the (d k)-volume of the typical cylinder base possesses a finite exponential moment. These estimates enable us to apply the celebrated Lemma on large deviations of Statuleviˇ cius. MSC: primary 60D05, 60F05; secondary 60F10, 60G55