BERRY-ESSEEN BOUNDS AND CRAMÉR-TYPE LARGE DEVIATIONS FOR THE VOLUME DISTRIBUTION OF POISSON CYLINDER PROCESSES

A stationary Poisson cylinder process Π cyl (d,k) is composed of a stationary Poisson process of k-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 the d-volume V ϱ (d,k) of the union set of Π cyl (d,k) that covers ϱW as the scaling factor ϱ 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 ϱ (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 ϱ (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 Statulevicius.