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.
In this work the ℓ q -norms of points chosen uniformly at random in a centered regular simplex in high dimensions are studied. Berry-Esseen bounds in the regime 1 ≤ q < ∞ are derived and complemented by a non-central limit theorem together with moderate and large deviations in the case where q = ∞. A comparison with corresponding results for ℓ n p -balls is carried out as well.
In this work the
$\ell_q$
-norms of points chosen uniformly at random in a centered regular simplex in high dimensions are studied. Berry–Esseen bounds in the regime
$1\leq q < \infty$
are derived and complemented by a non-central limit theorem together with moderate and large deviations in the case where
$q=\infty$
. An application to the intersection volume of a regular simplex with an
$\ell_p^n$
-ball is also carried out.
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