A strong law of large numbers for stationary point processes

1979 ◽  
Vol 46 (2) ◽  
pp. 159-164 ◽  
Author(s):  
R. M. Cranwell ◽  
N. A. Weiss
Keyword(s):  
Stationary Point ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers ◽  
Stationary Point Processes

scholarly journals Continuum AB percolation and AB random geometric graphs

2014 ◽  
Vol 51 (A) ◽  
pp. 333-344 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Point Processes ◽  
Law Of Large Numbers ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Strong Law ◽  
Random Geometric Graphs ◽  
Large Numbers ◽  
The One ◽  
Distance Parameter

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

scholarly journals Continuum AB percolation and AB random geometric graphs

2014 ◽  
Vol 51 (A) ◽  
pp. 333-344 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Point Processes ◽  
Law Of Large Numbers ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Strong Law ◽  
Random Geometric Graphs ◽  
Large Numbers ◽  
The One ◽  
Distance Parameter

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

scholarly journals On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

On projected thick sections of stationary point processes and Boolean models

1996 ◽  
Vol 28 (2) ◽  
pp. 335-335
Author(s):  
Markus Kiderlen
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Linear Subspace ◽  
Isotropic Case ◽  
Thick Section ◽  
Boolean Models ◽  
Dimensional Linear Subspace ◽  
Stationary Point Process ◽  
Stationary Point Processes

For a stationary point process X of convex particles in ℝd the projected thick section process X(L) on a q-dimensional linear subspace L is considered. Formulae connecting geometric functionals, e.g. the quermass densities of X and X(L), are presented. They generalize the classical results of Miles (1976) and Davy (1976) which hold only in the isotropic case.

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