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 (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 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.

Stopping sets: Gamma-type results and hitting properties

1999 ◽  
Vol 31 (2) ◽  
pp. 355-366 ◽  
Author(s):  
Sergei Zuyev
Keyword(s):  
Point Process ◽  
Sampling Theorem ◽  
Random Sets ◽  
Scaling Properties ◽  
Stopping Sets ◽  
Stopping Set ◽  
Alternative Description ◽  
Natural Filtration ◽  
Random Figure ◽  
Optional Sampling Theorem

Recently in the paper by Møller and Zuyev (1996), the following Gamma-type result was established. Given n points of a homogeneous Poisson process defining a random figure, its volume is Γ(n,λ) distributed, where λ is the intensity of the process. In this paper we give an alternative description of the class of random sets for which the Gamma-type results hold. We show that it corresponds to the class of stopping sets with respect to the natural filtration of the point process with certain scaling properties. The proof uses the martingale technique for directed processes, in particular, an analogue of Doob's optional sampling theorem proved in Kurtz (1980). As well as being compact, this approach provides a new insight into the nature of geometrical objects constructed with respect to a Poisson point process. We show, in particular, that in this framework the probability that a point is covered by a stopping set does not depend on whether it is a point of the process or not.

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (03) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k } are the points of a renewal process and {Ak } are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

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.

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (3) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k} are the points of a renewal process and {Ak} are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

scholarly journals Joint Distribution of the Number of Vertices and the Area of Convex Hulls Generated by a Uniform Distribution in a Convex Polygon

2021 ◽  
pp. 232-243
Author(s):  
Isakjan M. Khamdamov ◽  
Zoya S. Chay
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convex Hull ◽  
Uniform Distribution ◽  
Joint Distribution ◽  
Poisson Point Process ◽  
Convex Polygon ◽  
Point Processes ◽  
Poisson Approximation ◽  
Convex Hulls

A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process

The distribution of the convex hull of a Gaussian sample

1980 ◽  
Vol 17 (03) ◽  
pp. 686-695 ◽  
Author(s):  
William F. Eddy
Keyword(s):  
Weak Convergence ◽  
Convex Hull ◽  
Poisson Point Process ◽  
Point Processes ◽  
Convex Sets ◽  
Continuous Mapping ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Limit Process ◽  
Gaussian Sample

The distribution of the convex hull of a random sample ofd-dimensional variables is described by embedding the collection of convex sets into the space of continuous functions on the unit sphere. Weak convergence of the normalized convex hull of a circular Gaussian sample to a process with extreme-value marginal distributions is demonstrated. The proof shows that an underlying sequence of point processes converges to a Poisson point process and then applies the continuous mapping theorem. Several properties of the limit process are determined.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (4) ◽  
pp. 732-739 ◽  
Author(s):  
Bruce M. Brown ◽  
Sidney I. Resnick
Keyword(s):  
Stochastic Processes ◽  
Point Process ◽  
Time Scales ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extreme Values ◽  
Weak Limit ◽  
One Dimensional ◽  
Limit Process ◽  
Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

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