We study point processes that consist of certain centers of point tuples of an underlying Poisson process. Such processes can be used in stochastic geometry to study exceedances of various functionals describing geometric properties of the Poisson process. Using a coupling of the point process with its Palm version we prove a general Poisson limit theorem. We then apply our theorem to find the asymptotic distribution of the maximal volume content of random k-nearest neighbor balls. Combining our general result with the theory of asymptotic shapes of large cells in random mosaics, we prove a Poisson limit theorem for cell centers in the Poisson-Voronoi and -Delaunay mosaic. As a consequence, we establish Gumbel limits for the asymptotic distribution of the maximal cell size in the Poisson-Voronoi and -Delaunay mosaic w.r.t. a general size functional.
Recursive max-linear structural equation models with regularly varying noise variables are considered. Their causal structure is represented by a directed acyclic graph (DAG). The problem of identifying a recursive max-linear model and its associated DAG from its matrix of pairwise tail dependence coefficients is discussed. For example, it is shown that if a causal ordering of the associated DAG is additionally known, then the minimum DAG representing the recursive structural equations can be recovered from the tail dependence matrix. For the relevant subclass of recursive max-linear models, identifiability of the associated minimum DAG from the tail dependence matrix and the initial nodes is shown. Algorithms find the associated minimum DAG for the different situations. Furthermore, given a tail dependence matrix, an algorithm outputs all compatible recursive max-linear models and their associated minimum DAGs.
Let
$X_1,X_2, \ldots, X_n$
be a sequence of independent random points in
$\mathbb{R}^d$
with common Lebesgue density f. Under some conditions on f, we obtain a Poisson limit theorem, as
$n \to \infty$
, for the number of large probability kth-nearest neighbor balls of
$X_1,\ldots, X_n$
. Our result generalizes Theorem 2.2 of [11], which refers to the special case
$k=1$
. Our proof is completely different since it employs the Chen–Stein method instead of the method of moments. Moreover, we obtain a rate of convergence for the Poisson approximation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.