Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain

1995 ◽  
Vol 47 (1) ◽  
pp. 21-29 ◽  
Author(s):  
A. V. Nagaev
Keyword(s):  
Convex Domain ◽  
Point Processes ◽  
Convex Hulls ◽  
Poisson Point Processes ◽  
Unbounded Convex ◽  
Unbounded Convex Domain

Laplace transform identities for the volume of stopping sets based on Poisson point processes

2015 ◽  
Vol 47 (4) ◽  
pp. 919-933 ◽  
Author(s):  
Nicolas Privault
Keyword(s):  
Finite Difference ◽  
Laplace Transform ◽  
Finite Volume ◽  
Volume Content ◽  
Point Processes ◽  
Partial Ordering ◽  
Convex Hulls ◽  
Poisson Point Processes ◽  
Stopping Sets ◽  
Index Sets

We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on anticipating Girsanov identities for Poisson point processes under a cyclic vanishing condition for a finite difference gradient. This approach does not require classical assumptions based on set-indexed martingales and the (partial) ordering of index sets. The examples treated focus on stopping sets in finite volume, and include the random missed volume of Poisson convex hulls.

Laplace transform identities for the volume of stopping sets based on Poisson point processes

2015 ◽  
Vol 47 (04) ◽  
pp. 919-933
Author(s):  
Nicolas Privault
Keyword(s):  
Finite Difference ◽  
Laplace Transform ◽  
Finite Volume ◽  
Volume Content ◽  
Point Processes ◽  
Partial Ordering ◽  
Convex Hulls ◽  
Poisson Point Processes ◽  
Stopping Sets ◽  
Index Sets

We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on anticipating Girsanov identities for Poisson point processes under a cyclic vanishing condition for a finite difference gradient. This approach does not require classical assumptions based on set-indexed martingales and the (partial) ordering of index sets. The examples treated focus on stopping sets in finite volume, and include the random missed volume of Poisson convex hulls.

scholarly journals Cones generated by random points on half-spheres and convex hulls of Poisson point processes

2019 ◽  
Vol 175 (3-4) ◽  
pp. 1021-1061 ◽  
Author(s):  
Zakhar Kabluchko ◽  
Alexander Marynych ◽  
Daniel Temesvari ◽  
Christoph Thäle
Keyword(s):  
Point Processes ◽  
Convex Hulls ◽  
Poisson Point Processes

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

scholarly journals A generalization of Matérn hard-core processes with applications to max-stable processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1298-1312
Author(s):  
Martin Dirrler ◽  
Christopher Dörr ◽  
Martin Schlather
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Hard Core ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Poisson Point Processes ◽  
Original Process ◽  
Mixed Moving Maxima

AbstractMatérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process.

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