scholarly journals On Limit Theorem for the Number of Vertices of the Convex Hulls in a Unit Disk

2020 ◽  
pp. 275-284
Author(s):  
Isakjan M. Khamdamov
Keyword(s):  
Limit Theorem ◽  
Unit Disk ◽  
Point Process ◽  
Poisson Point Process ◽  
Convex Hulls ◽  
Strong Mixing ◽  
The Central Limit Theorem ◽  
Asymptotic Expressions ◽  
Independent Observations ◽  
Binomial Point Process

This paper is devoted to further investigation of the property of a number of vertices of convex hulls generated by independent observations of a two-dimensional random vector with regular distributions near the boundary of support when it is a unit disk. Following P. Groeneboom [4], the Binomial point process is approximated by the Poisson point process near the boundary of support and vertex processes of convex hulls are constructed. The properties of strong mixing and martingality of vertex processes are investigated. Using these properties, asymptotic expressions are obtained for the expectations and variance of the vertex processes that correspond to the results previously obtained by H. Carnal [2]. Further, using the properties of strong mixing of vertex processes, the central limit theorem for a number of vertices of a convex hull is proved

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 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

CLTs for Dependent Processes

2021 ◽  
pp. 548-567
Author(s):  
James Davidson
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Strong Mixing ◽  
Mixing Processes ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Law ◽  
Dependent Processes

This chapter deals with the central limit theorem (CLT) for dependent processes. As with the law of large numbers, the focus is on near‐epoch dependent and mixing processes and array versions of the results are given to allow heterogeneity. The cornerstone of these results is a general CLT due to McLeish, from which a result for martingales is obtained directly. A result for stationary ergodic mixingales is given, and the rest of the chapter is devoted to proving and interpreting a CLT for mixingales and hence for arrays that are near‐epoch dependent on a strong‐mixing and uniform-mixing processes.

Focusing of the scan statistic and geometric clique number

2002 ◽  
Vol 34 (4) ◽  
pp. 739-753 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Poisson Point Process ◽  
Dimensional Space ◽  
Clique Number ◽  
Geometric Graph ◽  
Finite Range ◽  
Scan Statistic ◽  
Constant Intensity ◽  
Binomial Point Process

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

A Note on Serial Correlation Coefficients and Related Characterizations of Gamma and Exponential Distributions

1972 ◽  
Vol 21 (1-2) ◽  
pp. 57-62 ◽  
Author(s):  
D. N. Shanbhag ◽  
I. V. Basawa
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Correlation Coefficient ◽  
Serial Correlation ◽  
Correlation Coefficients ◽  
Exponential Distributions ◽  
The Central Limit Theorem ◽  
Independent Observations ◽  
Serial Correlation Coefficient

Summary Some properties of the distribution of the serial correlation coefficient based on a sample of independent observations and that of a certain related statistic are used to characterize gamma and exponential distributions. Also, a certain extension of the central limit theorem is given which may be useful in establishing the asymptotic normality of serial correla­ tion­type statistics.

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.

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