Long-range dependent point processes and their Palm-Khinchin distributions

2000 ◽  
Vol 32 (4) ◽  
pp. 1051-1063 ◽  
Author(s):  
D. J. Daley ◽  
T. Rolski ◽  
Rein Vesilo
Keyword(s):  
Point Process ◽  
Long Range ◽  
Stationary Point ◽  
Point Processes ◽  
Interval Length ◽  
Hurst Index ◽  
Stationary Point Process ◽  
Stationary Interval ◽  
Range Interval ◽  
Dependent Point

For a stationary long-range dependent point process N(.) with Palm distribution P0, the Hurst index H ≡ sup{h : lim sup t→∞t-2h var N(0,t] = ∞} is related to the moment index κ ≡ sup{k : E0(Tk) < ∞} of a generic stationary interval T between points (E0 denotes expectation with respect to P0) by 2H + κ ≥ 3, it being known that equality holds for a stationary renewal process. Thus, a stationary point process for which κ < 2 is necessarily long-range dependent with Hurst index greater than ½. An extended example of a Wold process shows that a stationary point process can be both long-range count dependent and long-range interval dependent and have finite mean square interval length, i.e., E0(T2) < ∞.

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.

Some multivariate generalizations of results in univariate stationary point processes

1977 ◽  
Vol 14 (04) ◽  
pp. 748-757 ◽  
Author(s):  
Mark Berman
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Stationary Point Process ◽  
Stationary Point Processes

Some relationships are derived between the asynchronous and partially synchronous counting and interval processes associated with a multivariate stationary point process. A few examples are given to illustrate some of these relationships.

On a class of random motions of point processes involving interaction

1978 ◽  
Vol 10 (3) ◽  
pp. 613-632 ◽  
Author(s):  
Harry M. Pierson
Keyword(s):  
Large Class ◽  
Uniform Distribution ◽  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
General Setting ◽  
Invariant Distributions ◽  
Stationary Point Process ◽  
Random Motions ◽  
Interval Lengths

Starting with a stationary point process on the line with points one unit apart, simultaneously replace each point by a point located uniformly between the original point and its right-hand neighbor. Iterating this transformation, we obtain convergence to a limiting point process, which we are able to identify. The example of the uniform distribution is for purposes of illustration only; in fact, convergence is obtained for almost any distribution on [0, 1]. In the more general setting, we prove the limiting distribution is invariant under the above transformation, and that for each such transformation, a large class of initial processes leads to the same invariant distribution. We also examine the covariance of the limiting sequence of interval lengths. Finally, we identify those invariant distributions with independent interval lengths, and the transformations from which they arise.

scholarly journals A Remark on the Van Lieshout and Baddeley J-Function for Point Processes

1997 ◽  
Vol 29 (1) ◽  
pp. 19-25 ◽  
Author(s):  
T. Bedford ◽  
J. Van Den Berg
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Spatial Interaction ◽  
Empty Space ◽  
Stationary Point Process ◽  
Space Function ◽  
The One ◽  
Classical Construction

The empty space function of a stationary point process in ℝd is the function that assigns to each r, r > 0, the probability that there is no point within distance r of O. In a recent paper Van Lieshout and Baddeley study the so-called J-function, which is defined as the ratio of the empty space function of a stationary point process and that of its corresponding reduced Palm process. They advocate the use of the J-function as a characterization of the type of spatial interaction.Therefore it is natural to ask whether J ≡ 1 implies that the point process is Poisson. We restrict our analysis to the one-dimensional case and show that a classical construction by Szász provides an immediate counterexample. In this example the interpoint distances are still exponentially distributed. This raises the question whether it is possible to have J ≡ 1 but non-exponentially distributed interpoint distances. We construct a point process with J ≡ 1 but where the interpoint distances are bounded.

Weak homogenization of point processes by space deformations

2000 ◽  
Vol 32 (4) ◽  
pp. 948-959 ◽  
Author(s):  
R. Senoussi ◽  
J. Chadœuf ◽  
D. Allard
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Second Order ◽  
Poisson Processes ◽  
Borel Set ◽  
First Order ◽  
Stationary Point Process

We study the transformation of a non-stationary point process ξ on ℝn into a weakly stationary point process ͂ξ, with ͂ξ(B) = ξ(Φ-1(B)), where B is a Borel set, via a deformation Φ of the space ℝn. When the second-order measure is regular, Φ is uniquely determined by the homogenization equations of the second-order measure. In contrast, the first-order homogenization transformation is not unique. Several examples of point processes and transformations are investigated with a particular interest to Poisson processes.

scholarly journals A Remark on the Van Lieshout and Baddeley J-Function for Point Processes

1997 ◽  
Vol 29 (01) ◽  
pp. 19-25 ◽  
Author(s):  
T. Bedford ◽  
J. Van Den Berg
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Spatial Interaction ◽  
Empty Space ◽  
Stationary Point Process ◽  
Space Function ◽  
The One ◽  
Classical Construction

The empty space function of a stationary point process in ℝd is the function that assigns to each r, r &gt; 0, the probability that there is no point within distance r of O. In a recent paper Van Lieshout and Baddeley study the so-called J-function, which is defined as the ratio of the empty space function of a stationary point process and that of its corresponding reduced Palm process. They advocate the use of the J-function as a characterization of the type of spatial interaction. Therefore it is natural to ask whether J ≡ 1 implies that the point process is Poisson. We restrict our analysis to the one-dimensional case and show that a classical construction by Szász provides an immediate counterexample. In this example the interpoint distances are still exponentially distributed. This raises the question whether it is possible to have J ≡ 1 but non-exponentially distributed interpoint distances. We construct a point process with J ≡ 1 but where the interpoint distances are bounded.

On projected thick sections of stationary point processes and Boolean models

1996 ◽  
Vol 28 (02) ◽  
pp. 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 ℝ d 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.

Ergodicity and identifiability for random translations of stationary point processes

1975 ◽  
Vol 12 (04) ◽  
pp. 734-743
Author(s):  
Toshio Mori
Keyword(s):  
Point Process ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Identification Problem ◽  
Original Process ◽  
The Real ◽  
Displacement Distribution ◽  
Stationary Point Process ◽  
Stationary Point Processes

A bivariate point process consisting of an original stationary point process and its random translation is considered. Westcott's method is applied to show that if the original point process is ergodic then the bivariate point process is also ergodic. This result is applied to an identification problem of the displacement distribution. It is shown that if the spectrum of the original process is the real line then the displacement distribution is identifiable from almost every sample realisation of the bivariate process.

Some multivariate generalizations of results in univariate stationary point processes

1977 ◽  
Vol 14 (4) ◽  
pp. 748-757 ◽  
Author(s):  
Mark Berman
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Stationary Point Process ◽  
Stationary Point Processes

Some relationships are derived between the asynchronous and partially synchronous counting and interval processes associated with a multivariate stationary point process. A few examples are given to illustrate some of these relationships.

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