On the coverage of space by random sets

2004 ◽  
Vol 36 (01) ◽  
pp. 1-18 ◽  
Author(s):  
Siva Athreya ◽  
Rahul Roy ◽  
Anish Sarkar
Keyword(s):  
Markov Chain ◽  
Positive Integer ◽  
Point Process ◽  
Poisson Point Process ◽  
Random Variables ◽  
Random Sets ◽  
Random Set ◽  
Complete Coverage ◽  
Boolean Models

Let ξ1, ξ2,… be a Poisson point process of density λ on (0,∞) d , d ≥ 1, and let ρ, ρ1, ρ2,… be i.i.d. positive random variables independent of the point process. Let C := ⋃ i≥1 {ξ i + [0,ρ i ] d }. If, for some t > 0, (0,∞) d ⊆ C, then we say that (0,∞) d is eventually covered by C. We show that the eventual coverage of (0,∞) d depends on the behaviour of xP(ρ > x) as x → ∞ as well as on whether d = 1 or d ≥ 2. These results may be compared to those known for complete coverage of ℝ d by such Poisson Boolean models. In addition, we consider the set ⋃{i≥1:X i =1} [i,i+ρ i ], where X 1, X 2,… is a {0,1}-valued Markov chain and ρ1, ρ2,… are i.i.d. positive-integer-valued random variables independent of the Markov chain. We study the eventual coverage properties of this random set.

On the coverage of space by random sets

2004 ◽  
Vol 36 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Siva Athreya ◽  
Rahul Roy ◽  
Anish Sarkar
Keyword(s):  
Markov Chain ◽  
Positive Integer ◽  
Point Process ◽  
Poisson Point Process ◽  
Random Variables ◽  
Random Sets ◽  
Random Set ◽  
Complete Coverage ◽  
Boolean Models

Let ξ1, ξ2,… be a Poisson point process of density λ on (0,∞)d, d ≥ 1, and let ρ, ρ1, ρ2,… be i.i.d. positive random variables independent of the point process. Let C := ⋃i≥1 {ξi + [0,ρi]d}. If, for some t > 0, (0,∞)d ⊆ C, then we say that (0,∞)d is eventually covered by C. We show that the eventual coverage of (0,∞)d depends on the behaviour of xP(ρ > x) as x → ∞ as well as on whether d = 1 or d ≥ 2. These results may be compared to those known for complete coverage of ℝd by such Poisson Boolean models. In addition, we consider the set ⋃{i≥1:Xi=1} [i,i+ρi], where X1, X2,… is a {0,1}-valued Markov chain and ρ1, ρ2,… are i.i.d. positive-integer-valued random variables independent of the Markov chain. We study the eventual coverage properties of this random set.

Variance reducing modifications for estimators of standardized moments of random sets

2000 ◽  
Vol 32 (03) ◽  
pp. 682-700
Author(s):  
Jeffrey D. Picka
Keyword(s):  
Statistical Analysis ◽  
Point Process ◽  
Random Sets ◽  
Boolean Models ◽  
Asymptotic Regime ◽  
Estimation Strategy ◽  
Moment Estimation ◽  
Second Moments ◽  
Simulation Results ◽  
Variance Result

In the statistical analysis of random sets, it is useful to have simple statistics that can be used to describe the realizations of these sets. The cumulants and several other standardized moments such as the correlation and second cumulant can be used for this purpose, but their estimators can be excessively variable if the most straightforward estimation strategy is used. Through exploitation of similarities between this estimation problem and a similar one for a point process statistic, two modifications are proposed. Analytical results concerning the effects of these modifications are found through use of a specialized asymptotic regime. Simulation results establish that the modifications are highly effective at reducing estimator standard deviations for Boolean models. The results suggest that the reductions in variance result from a balanced use of information in the estimation of the first and second moments, through eliminating the use of observations that are not used in second moment estimation.

scholarly journals On Laslett's transform for the Boolean model

2001 ◽  
Vol 33 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A. D. Barbour ◽  
V. Schmidt
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Simple Proof ◽  
Poisson Point Process ◽  
Boolean Model ◽  
Random Sets ◽  
Convex Compact ◽  
Homogeneous Poisson Process ◽  
Vertical Strip ◽  
Cumulative Process

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

scholarly journals Asymptotics of geometrical navigation on a random set of points in the plane

2011 ◽  
Vol 43 (4) ◽  
pp. 899-942 ◽  
Author(s):  
Nicolas Bonichon ◽  
Jean-François Marckert
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Finite Domain ◽  
Random Set ◽  
Asymptotic Results ◽  
Path Lengths ◽  
Set Of Points ◽  
Geometric Nature

A navigation on a set of points S is a rule for choosing which point to move to from the present point in order to progress toward a specified target. We study some navigations in the plane where S is a nonuniform Poisson point process (in a finite domain) with intensity going to +∞. We show the convergence of the traveller's path lengths, and give the number of stages and the geometry of the traveller's trajectories, uniformly for all starting points and targets, for several navigations of geometric nature. Other costs are also considered. This leads to asymptotic results on the stretch factors of random Yao graphs and random θ-graphs.

Variance reducing modifications for estimators of standardized moments of random sets

2000 ◽  
Vol 32 (3) ◽  
pp. 682-700 ◽  
Author(s):  
Jeffrey D. Picka
Keyword(s):  
Statistical Analysis ◽  
Point Process ◽  
Random Sets ◽  
Boolean Models ◽  
Asymptotic Regime ◽  
Estimation Strategy ◽  
Moment Estimation ◽  
Second Moments ◽  
Simulation Results ◽  
Variance Result

In the statistical analysis of random sets, it is useful to have simple statistics that can be used to describe the realizations of these sets. The cumulants and several other standardized moments such as the correlation and second cumulant can be used for this purpose, but their estimators can be excessively variable if the most straightforward estimation strategy is used. Through exploitation of similarities between this estimation problem and a similar one for a point process statistic, two modifications are proposed. Analytical results concerning the effects of these modifications are found through use of a specialized asymptotic regime. Simulation results establish that the modifications are highly effective at reducing estimator standard deviations for Boolean models. The results suggest that the reductions in variance result from a balanced use of information in the estimation of the first and second moments, through eliminating the use of observations that are not used in second moment estimation.

On the variance to mean ratio for random variables from Markov chains and point processes

1998 ◽  
Vol 35 (2) ◽  
pp. 303-312 ◽  
Author(s):  
Timothy C. Brown ◽  
Kais Hamza ◽  
Aihua Xia
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Point Process ◽  
Continuous Time ◽  
Infinitesimal Generator ◽  
Point Processes ◽  
Random Variables ◽  
Simple Point ◽  
Conditional Intensity

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.

On the variance to mean ratio for random variables from Markov chains and point processes

1998 ◽  
Vol 35 (02) ◽  
pp. 303-312 ◽  
Author(s):  
Timothy C. Brown ◽  
Kais Hamza ◽  
Aihua Xia
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Point Process ◽  
Continuous Time ◽  
Infinitesimal Generator ◽  
Point Processes ◽  
Random Variables ◽  
Simple Point ◽  
Conditional Intensity

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.

scholarly journals Asymptotics of geometrical navigation on a random set of points in the plane

2011 ◽  
Vol 43 (04) ◽  
pp. 899-942 ◽  
Author(s):  
Nicolas Bonichon ◽  
Jean-François Marckert
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Finite Domain ◽  
Random Set ◽  
Asymptotic Results ◽  
Path Lengths ◽  
Set Of Points ◽  
Geometric Nature

A navigation on a set of pointsSis a rule for choosing which point to move to from the present point in order to progress toward a specified target. We study some navigations in the plane whereSis a nonuniform Poisson point process (in a finite domain) with intensity going to +∞. We show the convergence of the traveller's path lengths, and give the number of stages and the geometry of the traveller's trajectories, uniformly for all starting points and targets, for several navigations of geometric nature. Other costs are also considered. This leads to asymptotic results on the stretch factors of random Yao graphs and random θ-graphs.

Poisson approximation related to spectra of hierarchical Laplacians

2019 ◽  
Vol 20 (05) ◽  
pp. 2050035
Author(s):  
Alexander Bendikov ◽  
Wojciech Cygan
Keyword(s):  
Continuous Function ◽  
Point Process ◽  
Density Of States ◽  
Poisson Point Process ◽  
Point Processes ◽  
Point Spectrum ◽  
Random Variables ◽  
Poisson Approximation ◽  
Ultrametric Space ◽  
Locally Compact

Let [Formula: see text] be a locally compact separable ultrametric space. Given a measure [Formula: see text] on [Formula: see text] and a function [Formula: see text] defined on the set of all non-singleton balls [Formula: see text] of [Formula: see text], we consider the hierarchical Laplacian [Formula: see text]. The operator [Formula: see text] acts in [Formula: see text] is essentially self-adjoint and has a purely point spectrum. Choosing a sequence [Formula: see text] of i.i.d. random variables, we consider the perturbed function [Formula: see text] and the perturbed hierarchical Laplacian [Formula: see text] Under certain conditions, the density of states [Formula: see text] exists and it is a continuous function. We choose a point [Formula: see text] such that [Formula: see text] and build a sequence of point processes defined by the eigenvalues of [Formula: see text] located in the vicinity of [Formula: see text]. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity [Formula: see text].

scholarly journals On the least common multiple of random q-integers

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Probabilistic Model ◽  
Expected Value ◽  
Random Set ◽  
Common Multiple ◽  
Least Common Multiple

AbstractFor every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.

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