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.

Equality of critical densities in continuum percolation

1995 ◽  
Vol 32 (01) ◽  
pp. 90-104 ◽  
Author(s):  
Ronald Meester
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Euclidean Distance ◽  
Integrability Condition ◽  
Continuum Percolation ◽  
Positive Probability ◽  
Homogeneous Poisson Process ◽  
Extra Point

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

scholarly journals Geometry of the Poisson Boolean model on a region of logarithmic width in the plane

2011 ◽  
Vol 43 (03) ◽  
pp. 616-635
Author(s):  
Amites Dasgupta ◽  
Rahul Roy ◽  
Anish Sarkar
Keyword(s):  
Phase Transition ◽  
Half Space ◽  
Point Process ◽  
Poisson Point Process ◽  
Random Variable ◽  
Boolean Model ◽  
Side Length ◽  
Tail Behaviour ◽  
Critical Intensity ◽  
Poisson Boolean Model

Consider the region L = {(x, y): 0 ≤ y ≤ Clog(1 + x), x > 0} for a constant C > 0. We study the percolation and coverage properties of this region. For the coverage properties, we place a Poisson point process of intensity λ on the entire half space R + x R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L. For the percolation properties, we place a Poisson point process of intensity λ on the region R 2. At each point of the process we centre a box of a random side length ρ. In the case ρ ≤ R for some fixed R > 0 we study the critical intensity λc of the percolation on L.

scholarly journals Poisson approximation for some point processes in reliability

2004 ◽  
Vol 36 (2) ◽  
pp. 455-470 ◽  
Author(s):  
Jean-Bernard Gravereaux ◽  
James Ledoux
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Poisson Approximation ◽  
Arrival Process ◽  
Homogeneous Poisson Process ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Finite Markov Process ◽  
Failure Point

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

scholarly journals Poisson approximation for some point processes in reliability

2004 ◽  
Vol 36 (02) ◽  
pp. 455-470
Author(s):  
Jean-Bernard Gravereaux ◽  
James Ledoux
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Poisson Approximation ◽  
Arrival Process ◽  
Homogeneous Poisson Process ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Finite Markov Process ◽  
Failure Point

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

scholarly journals Estimation of the local intensity of a cyclic Poisson process by means of nearest neighbor distances

2002 ◽  
Vol 1 (1) ◽  
pp. 1
Author(s):  
I W. MANGKU
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Poisson Point Process ◽  
Nearest Neighbor ◽  
Single Realization ◽  
Local Intensity ◽  
Strongly Consistent ◽  
Nearest Neighbor Distances

We consider the problem of estimating the local intensity of a cyclic Poisson point process, when we know the period. We suppose that only a single realization of the cyclic Poisson point process is observed within a bounded 'window', and our aim is to estimate consistently the local intensity at a given point. A nearest neighbor estimator of the local intensity is proposed, and we show that our estimator is weakly and strongly consistent, as the window expands.

Properties of the bivariate delayed Poisson process

1975 ◽  
Vol 12 (02) ◽  
pp. 257-268 ◽  
Author(s):  
A. J. Lawrance ◽  
P. A. W. Lewis
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Joint Distribution ◽  
Poisson Point Process ◽  
Initial Conditions ◽  
Arbitrary Time ◽  
Delay Distribution ◽  
Counting Distribution

The bivariate Poisson point process introduced in Cox and Lewis (1972), and there called the bivariate delayed Poisson process, is studied further; the process arises from pairs of delays on the events of a Poisson process. In particular, results are obtained for the stationary initial conditions, the joint distribution of the number of operative delays at an arbitrary time, the asynchronous counting distribution, and two semi-synchronous interval distributions. The joint delay distribution employed allows for dependence and two-sided delays, but the model retains the independence between different pairs of delays.

Equality of critical densities in continuum percolation

1995 ◽  
Vol 32 (1) ◽  
pp. 90-104 ◽  
Author(s):  
Ronald Meester
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Euclidean Distance ◽  
Integrability Condition ◽  
Continuum Percolation ◽  
Positive Probability ◽  
Homogeneous Poisson Process ◽  
Extra Point

Consider a homogeneous Poisson process inwith density ρ, and add the origin as an extra point. Now connect any two pointsxandyof the process with probabilityg(x − y), independently of the point process and all other pairs, wheregis a function which depends only on the Euclidean distance betweenxandy, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if, then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that ifgis of the form, for somer > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

scholarly journals Geometry of the Poisson Boolean model on a region of logarithmic width in the plane

2011 ◽  
Vol 43 (3) ◽  
pp. 616-635
Author(s):  
Amites Dasgupta ◽  
Rahul Roy ◽  
Anish Sarkar
Keyword(s):  
Phase Transition ◽  
Half Space ◽  
Point Process ◽  
Poisson Point Process ◽  
Random Variable ◽  
Boolean Model ◽  
Side Length ◽  
Tail Behaviour ◽  
Critical Intensity ◽  
Poisson Boolean Model

Consider the region L = {(x, y): 0 ≤ y ≤ Clog(1 + x), x > 0} for a constant C > 0. We study the percolation and coverage properties of this region. For the coverage properties, we place a Poisson point process of intensity λ on the entire half space R+ x R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L. For the percolation properties, we place a Poisson point process of intensity λ on the region R2. At each point of the process we centre a box of a random side length ρ. In the case ρ ≤ R for some fixed R > 0 we study the critical intensity λc of the percolation on L.

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.

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.

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