scholarly journals Asymptotics for Some Combinatorial Characteristics of the Convex Hull of a Poisson Point Process in the Clifford Torus

2012 ◽  
Vol 49 (2) ◽  
pp. 200-220
Author(s):  
Alexander Magazinov
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Poisson Point Process ◽  
Clifford Torus

Convergence in mean of some characteristics of the convex hull

1989 ◽  
Vol 21 (3) ◽  
pp. 526-542 ◽  
Author(s):  
Henk Brozius
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Regular Variation ◽  
Poisson Point Process ◽  
Random Vectors ◽  
Area And Perimeter ◽  
Convergence In Mean ◽  
Quantities Of Interest ◽  
Independent And Identically Distributed

A sequence Xn, 1 of independent and identically distributed random vectors is considered. Under a condition of regular variation, the number of vertices of the convex hull of {X1, …, Xn} converges in distribution to the number of vertices of the convex hull of a certain Poisson point process. In this paper, it is proved without sharpening the conditions that the expectation of this number also converges; expressions are found for its limit, generalizing results of Davis et al. (1987). We also present some results concerning other quantities of interest, such as area and perimeter of the convex hull and the probability that a given point belongs to the convex hull.

scholarly journals On the peeling procedure applied to a Poisson point process

2010 ◽  
Vol 42 (3) ◽  
pp. 620-630
Author(s):  
Y. Davydov ◽  
A. Nagaev ◽  
A. Philippe
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Numerical Experiments ◽  
Poisson Point Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Convex Hulls ◽  
Limit Shape ◽  
Empirical Point ◽  
Stable Random Vectors

In this paper we focus on the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a larger class of Poisson point processes.

scholarly journals Cardinality estimation for random stopping sets based on Poisson point processes

2021 ◽  
Author(s):  
Nicolas Privault
Keyword(s):  
Convex Hull ◽  
Numerical Simulations ◽  
Poisson Distribution ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Unbiased Estimators ◽  
Stopping Sets ◽  
Stopping Set ◽  
Standard Sampling

We construct unbiased estimators for the distribution of the number of vertices inside random stopping sets based on a Poisson point process. Our approach is based on moment identities for stopping sets, showing that the random count of points inside the complement of a stopping set S has a Poisson distribution conditionally to S. The proofs do not require the use of set-indexed martingales, and our estimators have a lower variance when compared to standard sampling. Numerical simulations are presented for examples such as the convex hull and the Voronoi flower of a Poisson point process, and their complements.

scholarly journals On the peeling procedure applied to a Poisson point process

2010 ◽  
Vol 42 (03) ◽  
pp. 620-630
Author(s):  
Y. Davydov ◽  
A. Nagaev ◽  
A. Philippe
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Numerical Experiments ◽  
Poisson Point Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Convex Hulls ◽  
Limit Shape ◽  
Empirical Point ◽  
Stable Random Vectors

In this paper we focus on the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a larger class of Poisson point processes.

Convergence in mean of some characteristics of the convex hull

1989 ◽  
Vol 21 (03) ◽  
pp. 526-542 ◽  
Author(s):  
Henk Brozius
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Regular Variation ◽  
Poisson Point Process ◽  
Random Vectors ◽  
Area And Perimeter ◽  
Convergence In Mean ◽  
Quantities Of Interest ◽  
Independent And Identically Distributed

A sequence X n , 1 of independent and identically distributed random vectors is considered. Under a condition of regular variation, the number of vertices of the convex hull of { X 1, …, X n} converges in distribution to the number of vertices of the convex hull of a certain Poisson point process. In this paper, it is proved without sharpening the conditions that the expectation of this number also converges; expressions are found for its limit, generalizing results of Davis et al. (1987). We also present some results concerning other quantities of interest, such as area and perimeter of the convex hull and the probability that a given point belongs to the convex hull.

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