The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (1) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (01) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

On point processes on the circle

1986 ◽  
Vol 23 (02) ◽  
pp. 322-331
Author(s):  
Jürg Hüsler
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convergence Result ◽  
General Point ◽  
Coverage Problem ◽  
Fixed Length ◽  
General Limit ◽  
Limit Result

Point processes on the circle with circumference 1 are considered, which are related to the coverage problem of the circle by n randomly placed arcs of a fixed length. The anticlockwise endpoint of each arc is assumed to be uniformly distributed on the circle. We deal with a general limit result on the convergence of these point processes to a Poisson process on the circle. This result is then applied to several cases of the coverage problem, giving improved limit results in these cases. The proof uses a new convergence result of general point processes.

An order statistic characterization of the poisson renewal process

1985 ◽  
Vol 22 (3) ◽  
pp. 717-722 ◽  
Author(s):  
Uri Liberman
Keyword(s):  
Poisson Process ◽  
Order Statistic ◽  
Renewal Process ◽  
Point Processes

Using the characterization of point processes having the order statistic property we prove that the only renewal process that has the order statistic property is the Poisson process.

Some models for rainfall based on stochastic point processes

1987 ◽  
Vol 410 (1839) ◽  
pp. 269-288 ◽  
Keyword(s):  
Poisson Process ◽  
Stochastic Models ◽  
Point Processes ◽  
Rainfall Intensity ◽  
Storm Events ◽  
Total Rainfall ◽  
Random Intensity ◽  
Hourly Rainfall ◽  
Complex Models ◽  
Stochastic Point Processes

Stochastic models are discussed for the variation of rainfall intensity at a fixed point in space. First, models are analysed in which storm events arise in a Poisson process, each such event being associated with a period of rainfall of random duration and constant but random intensity. Total rainfall intensity is formed by adding the contributions from all storm events. Then similar but more complex models are studied in which storms arise in a Poisson process, each storm giving rise to a cluster of rain cells and each cell being associated with a random period of rain. The main properties of these models are determined analytically. Analysis of some hourly rainfall data from Denver, Colorado shows the clustered models to be much the more satisfactory.

A simple proof of the multivariate random time change theorem for point processes

1988 ◽  
Vol 25 (01) ◽  
pp. 210-214 ◽  
Author(s):  
Timothy C. Brown ◽  
M. Gopalan Nair
Keyword(s):  
Poisson Process ◽  
Simple Proof ◽  
Point Processes ◽  
Time Change ◽  
Random Time ◽  
Random Time Change ◽  
Special Case

A simple proof of the multivariate random time change theorem of Meyer (1971) is given. This result includes Watanabe's (1964) characterization of the Poisson process; even in this special case the present proof is simpler than existing proofs.

Dynamical systems defined on point processes

1998 ◽  
Vol 35 (03) ◽  
pp. 581-588
Author(s):  
Laurence A. Baxter
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Point Processes ◽  
Linear Boltzmann Equation ◽  
Semidynamical System ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient ◽  
The Relationship

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.

A simple proof of the multivariate random time change theorem for point processes

1988 ◽  
Vol 25 (1) ◽  
pp. 210-214 ◽  
Author(s):  
Timothy C. Brown ◽  
M. Gopalan Nair
Keyword(s):  
Poisson Process ◽  
Simple Proof ◽  
Point Processes ◽  
Time Change ◽  
Random Time ◽  
Random Time Change ◽  
Special Case

A simple proof of the multivariate random time change theorem of Meyer (1971) is given. This result includes Watanabe's (1964) characterization of the Poisson process; even in this special case the present proof is simpler than existing proofs.

An order statistic characterization of the poisson renewal process

1985 ◽  
Vol 22 (03) ◽  
pp. 717-722
Author(s):  
Uri Liberman
Keyword(s):  
Poisson Process ◽  
Order Statistic ◽  
Renewal Process ◽  
Point Processes

Using the characterization of point processes having the order statistic property we prove that the only renewal process that has the order statistic property is the Poisson process.

On point processes on the circle

1986 ◽  
Vol 23 (2) ◽  
pp. 322-331 ◽  
Author(s):  
Jürg Hüsler
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convergence Result ◽  
General Point ◽  
Coverage Problem ◽  
Fixed Length ◽  
General Limit ◽  
Limit Result

Point processes on the circle with circumference 1 are considered, which are related to the coverage problem of the circle by n randomly placed arcs of a fixed length. The anticlockwise endpoint of each arc is assumed to be uniformly distributed on the circle. We deal with a general limit result on the convergence of these point processes to a Poisson process on the circle. This result is then applied to several cases of the coverage problem, giving improved limit results in these cases. The proof uses a new convergence result of general point processes.

Dynamical systems defined on point processes

1998 ◽  
Vol 35 (3) ◽  
pp. 581-588
Author(s):  
Laurence A. Baxter
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Point Processes ◽  
Linear Boltzmann Equation ◽  
Semidynamical System ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient ◽  
The Relationship

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.

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