Concentration inequalities for functionals of Poisson cylinder processes

Abstract

Random union sets Z associated with stationary Poisson processes of k-cylinders in R d are considered. Under general conditions on the typical cylinder base a concentration inequality for the volume of Z restricted to a compact window is derived. Assuming convexity of the typical cylinder base and isotropy of Z a concentration inequality for intrinsic volumes of arbitrary order is established. A number of special cases are discussed, for example the case when the cylinder bases arise from a random rotation of a fixed convex body. Also the situation of expanding windows is studied. Special attention is payed to the case k = 0, which corresponds to the classical Boolean model.

Concentration on Poisson spaces via modified $Φ$-Sobolev inequalities

“…To compare the result of Proposition 6. for all relevant values of t. In other words, the concentration bound from [4] is in this case always better than the one implied by Proposition 6.4 by at least a constant factor.…”

“…In the final Section 6 we discuss two novel applications of our general results to models from stochastic geometry, namely Poisson polytopes and Poisson cylinder processes. While concentration bounds for geometric functionals of Poisson cylinder models are known from [4] and concentration inequalities for random polytopes can be found in [20], some of the estimates we prove are new. This is especially the case for the so-called intrinsic volumes of Poisson polytopes with vertices chosen from the boundary of a smooth convex body for which no concentration inequalities can be found in the existing literature.…”

“…Our second application deals with the Poisson cylinder model for which a number of concentration properties for various geometric functionals have recently been studied in [4]. We refer to that paper and the literature cited therein for further references and background material.…”

“…Concentration properties for F around its mean have recently been studied in [4, Section 3] (and [8] for k = 0) based on concentration inequalities deduced from general covariance identities for Poisson functionals. The purpose of this section is to demonstrate that Corollary 5.3 can also be used to derive concentration bounds for F , where in addition to [4] we assume that the typical cylinder base has uniformly bounded volume. To formulate our result, let Q * the marginal distribution of Q on the C d−k -coordinate and let Ξ be a random element with distribution Q * , the so-called typical cylinder base.…”

“…Using general covariance identities for exponential functions of Poisson processes Gieringer and Last [8] were able to derive a new set of concentration inequalities. They were particularly useful to study concentration properties of geometric functionals associated with the Poisson Boolean model [8] or with so-called Poisson cylinder processes [4] considered in stochastic geometry. Finally, based on a general transportation inequality concentration bounds for so-called convex functionals of Poisson processes were proved by Gozlan, Herry and Peccati [9].…”