Selective interaction of a Poisson and renewal process: first-order stationary point results

1970 ◽  
Vol 7 (2) ◽  
pp. 359-372 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Stationary Point ◽  
Joint Distribution ◽  
Point Processes ◽  
First Order ◽  
Response Interval ◽  
Selective Interaction ◽  
Counting Distribution ◽  
Stationary Point Process ◽  
Equilibrium Process ◽  
Stationary Point Processes

The simple stationarity of a previously derived equilibrium process of responses in a renewal inhibited stationary point process is established by deriving the joint distribution of the number of responses in contiguous intervals in the process. For a renewal inhibited Poisson process the variancetime function of the process is obtained; the distribution of an arbitrary between-response interval and the synchronous counting distribution are also derived following analytic justification of the required results. These results strengthen earlier results in the theory of stationary point processes. Three other point processes arising from the interaction are briefly discussed.

Selective interaction of a Poisson and renewal process: first-order stationary point results

1970 ◽  
Vol 7 (02) ◽  
pp. 359-372 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Stationary Point ◽  
Joint Distribution ◽  
Point Processes ◽  
First Order ◽  
Response Interval ◽  
Selective Interaction ◽  
Counting Distribution ◽  
Stationary Point Process ◽  
Equilibrium Process ◽  
Stationary Point Processes

The simple stationarity of a previously derived equilibrium process of responses in a renewal inhibited stationary point process is established by deriving the joint distribution of the number of responses in contiguous intervals in the process. For a renewal inhibited Poisson process the variancetime function of the process is obtained; the distribution of an arbitrary between-response interval and the synchronous counting distribution are also derived following analytic justification of the required results. These results strengthen earlier results in the theory of stationary point processes. Three other point processes arising from the interaction are briefly discussed.

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.

Selective interaction of a poisson and renewal process: the dependency structure of the intervals between responses

1971 ◽  
Vol 8 (1) ◽  
pp. 170-183 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Poisson Process ◽  
Stationary Point ◽  
Joint Distribution ◽  
Initial Conditions ◽  
Response Process ◽  
New Approach ◽  
Dependency Structure ◽  
Joint Distributions ◽  
Selective Interaction ◽  
Stationary Point Process

This paper studies the dependency structure of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b)). A new approach to the intervals between events in a stationary point process, based on the idea of an average event, is introduced. Average event initial conditions (as opposed to equilibrium initial conditions previously determined) for the renewal inhibited Poisson process are obtained and event stationarity of the resulting response process is established. The joint distribution and correlation between pairs of contiguous synchronous intervals is obtained; further, the joint distribution of non-contiguous pairs of synchronous intervals is derived. Finally, the joint distributions of pairs of contiguous synchronous and asynchronous intervals are related, and a similar but more general stationary point result is conjectured.

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.

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.

Selective interaction of a stationary point process and a renewal process

1970 ◽  
Vol 7 (02) ◽  
pp. 483-489 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Explicit Expression ◽  
Point Process ◽  
Renewal Process ◽  
Stationary Point ◽  
Equilibrium Conditions ◽  
Selective Interaction ◽  
Counting Distribution ◽  
Stationary Point Process

The selective interaction of a stationary point process and a renewal process is studied. Equilibrium conditions for the resulting point process are given, the equilibrium counting distribution is derived, and an explicit expression for the rate of the process is determined.

Ergodicity and identifiability for random translations of stationary point processes

1975 ◽  
Vol 12 (4) ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document