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.

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.

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.

Sufficient conditions for long-range count dependence of stationary point processes on the real line

2001 ◽  
Vol 38 (2) ◽  
pp. 570-581 ◽  
Author(s):  
Rafał Kulik ◽  
Ryszard Szekli
Keyword(s):  
Long Range ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Second Moment ◽  
The Real ◽  
Palm Measure ◽  
Stationary Point Processes

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

Sufficient conditions for long-range count dependence of stationary point processes on the real line

2001 ◽  
Vol 38 (02) ◽  
pp. 570-581 ◽  
Author(s):  
Rafał Kulik ◽  
Ryszard Szekli
Keyword(s):  
Long Range ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Second Moment ◽  
The Real ◽  
Palm Measure ◽  
Stationary Point Processes

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

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.

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.

scholarly journals The probability generating functional

1972 ◽  
Vol 14 (4) ◽  
pp. 448-466 ◽  
Author(s):  
M. Westcott
Keyword(s):  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Generating Functional ◽  
The Real ◽  
The Subject

This paper is concerned with certain aspects of the theory and application of the probability generating functional (p.g.fl) of a point process on the real line. Interest in point processes has increased rapidly during the last decade and a number of different approaches to the subject have been expounded (see for example [6], [11], [15], [17], [20], [25], [27], [28]). It is hoped that the present development using the p.g.ff will calrify and unite some of these viewpoints and provide a useful tool for solution of particular problems.

Bivariate stationary point processes, fundamental relations and first recurrence times

1972 ◽  
Vol 4 (02) ◽  
pp. 296-317 ◽  
Author(s):  
T. K. M. Wisniewski
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Counting Process ◽  
Derived Relations ◽  
Recurrence Times ◽  
Different Types ◽  
First Recurrence ◽  
Stationary Point Processes ◽  
Event Counting

Various types of time and event sampling of a stationary and orderly bivariate point process are considered. Fundamental relations between inter-event intervals and the event counting process are derived. Relations between first forward recurrence times and their moments for different types of sampling are obtained.

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.

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