A characterization of the probability-generating functional for point processes on measurable spaces

1977 ◽  
Vol 9 (3) ◽  
pp. 442-442
Author(s):  
J. Mecke
Keyword(s):  
Point Processes ◽  
Generating Functional

An analysis of the Pólya point process

1983 ◽  
Vol 15 (01) ◽  
pp. 39-53 ◽  
Author(s):  
Ed Waymire ◽  
Vijay K. Gupta
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Critical Value ◽  
General Representation ◽  
Infinitely Divisible ◽  
Generating Functional ◽  
Representation Problem ◽  
Polya Process ◽  
Pólya Process ◽  
Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

A cluster process representation of a self-exciting process

1974 ◽  
Vol 11 (3) ◽  
pp. 493-503 ◽  
Author(s):  
Alan G. Hawkes ◽  
David Oakes
Keyword(s):  
Point Processes ◽  
Generating Functional ◽  
Cluster Process ◽  
Process Representation ◽  
Age Dependent ◽  
Birth Processes ◽  
Poisson Cluster ◽  
Cluster Processes

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.

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.

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.

Further results for Gauss-Poisson processes

1972 ◽  
Vol 4 (01) ◽  
pp. 151-176 ◽  
Author(s):  
R. K. Milne ◽  
M. Westcott
Keyword(s):  
Point Processes ◽  
Sufficient Conditions ◽  
Infinite Divisibility ◽  
Poisson Processes ◽  
Generating Functional ◽  
Basic Approach ◽  
New Class ◽  
Two Measures ◽  
Necessary And Sufficient ◽  
Statistical Results

Newman (1970) introduced an interesting new class of point processes which he called Gauss-Poisson. They are characterized, in the most general case, by two measures. We determine necessary and sufficient conditions on these measures for the resulting point process to be well defined, and proceed to a systematic study of its properties. These include stationarity, ergodicity, and infinite divisibility. We mention connections with other classes of point processes and some statistical results. Our basic approach is through the probability generating functional of the process.

Stochastic population processes in the theory of radiative transfer

1970 ◽  
Vol 7 (02) ◽  
pp. 272-290
Author(s):  
P. J. Brockwell
Keyword(s):  
Radiative Transfer ◽  
Transition Probability ◽  
Collision Integral ◽  
Time Interval ◽  
Reciprocity Principle ◽  
Initial State ◽  
First Passage ◽  
Generating Functional ◽  
Generalized State

Summary Starting from a characterization of radiative transfer in terms of a collision rate λ and a single-collision transition probability Ψ, we study the distribution of the generalized state ζ(t) of a radiation particle at time t conditional on a specified initial state at time t = 0. The generalized state is a vector consisting of the state ω(t) at time t and the states ω 1, ω 2, …, ω n of the particle immediately after the collisions it experiences in the time interval (0, t]. The variable ζ(t) takes values in a population space and can be studied conveniently with the aid of a certain generating functional G. The first-collision integral equation and the backward integro-differential equation for G are derived. Simultaneous consideration of the first-collision and last-collision equations lead to a generalized reciprocity principle for G. First-passage problems are also considered. Finally a number of illustrative examples are given.

Characterization of compensators for point processes on the plane

1990 ◽  
Vol 29 (3) ◽  
pp. 395-405 ◽  
Author(s):  
B. Gail Ivanoff ◽  
Ely Merzbacht
Keyword(s):  
Point Processes

scholarly journals A method for the identification and characterization of clusters of schools along the transect lines of fisheries-acoustic surveys

2003 ◽  
Vol 60 (4) ◽  
pp. 872-884 ◽  
Author(s):  
Pierre Petitgas
Keyword(s):  
Point Processes ◽  
Fish Stock ◽  
Small Range ◽  
One Dimensional ◽  
School Based ◽  
Aggregation Pattern ◽  
Acoustic Surveys ◽  
School Events ◽  
Identification And Characterization

Abstract The school-aggregation pattern (schools and clusters of schools) is presumed to play a significant role in determining pelagic fish-stock catchability. However, its analysis has seldom been undertaken because it requires field-behavioural data that is seldom available. Such information can now be obtained by analysing school-based data of fisheries-acoustic surveys. This paper proposes a method for doing so. The method allows for the identification of clusters of schools and the estimation of their parameters along one-dimensional, acoustic-survey transect lines. It is based on a spatial point-process approach that considers schools as point events occurring along the track sailed by a ship. More precisely, it is based on defining a maximum distance between schools in a cluster. This distance is chosen to optimize various criteria and in particular that of homogeneity concerning school location inside the clusters and school number per km. The algorithm is described and applied to a series of acoustic surveys carried out in the Bay of Biscay. The pertinence of the clusters obtained by the algorithm is evaluated by analysing which component of the spatial distribution of the schools corresponds to those clusters. This involves considering all the distances between school events and performing simulations of cluster point processes. The school clusters obtained by the proposed algorithm represent a small-range structure of a few kilometres when a longer-range structure of tens of kilometres was also present in the data.

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