An analysis of the Pólya point process

1983 ◽  
Vol 15 (1) ◽  
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.

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.

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.

Rarefactions of compound point processes

1984 ◽  
Vol 21 (04) ◽  
pp. 710-719
Author(s):  
Richard F. Serfozo
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Classical Result ◽  
Point Processes ◽  
Rare Events ◽  
Random Measure ◽  
Bernoulli Trials ◽  
Random Measures ◽  
Infinitely Divisible ◽  
Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

On a class of parabolic differential equations driven with stochastic point processes

1975 ◽  
Vol 12 (01) ◽  
pp. 98-106
Author(s):  
K. Gopalsamy ◽  
A. T. Bharucha-Reid
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Point Process ◽  
Point Processes ◽  
Boundary Value ◽  
Parabolic Differential Equation ◽  
Parabolic Differential Equations ◽  
Stochastic Point Process ◽  
Stochastic Point Processes

This paper is concerned with the solution of an initial and boundary value problem for a parabolic differential equation driven by a stochastic point process.

On a class of multivariate counting processes

2016 ◽  
Vol 48 (2) ◽  
pp. 443-462 ◽  
Author(s):  
Ji Hwan Cha ◽  
Massimiliano Giorgio
Keyword(s):  
Point Processes ◽  
Dependence Structure ◽  
Counting Processes ◽  
New Class ◽  
Marginal Process ◽  
Polya Process ◽  
History Of ◽  
Pólya Process ◽  
Important Characterization

Abstract In this paper we define and study a new class of multivariate counting processes, named `multivariate generalized Pólya process'. Initially, we define and study the bivariate generalized Pólya process and briefly discuss its reliability application. In order to derive the main properties of the process, we suggest some key properties and an important characterization of the process. Due to these properties and the characterization, the main properties of the bivariate generalized Pólya process are obtained efficiently. The marginal processes of the multivariate generalized Pólya process are shown to be the univariate generalized Pólya processes studied in Cha (2014). Given the history of a marginal process, the conditional property of the other process is also discussed. The bivariate generalized Pólya process is extended to the multivariate case. We define a new dependence concept for multivariate point processes and, based on it, we analyze the dependence structure of the multivariate generalized Pólya process.

NONPARAMETRIC DETECTION OF DEPENDENCES IN STOCHASTIC POINT PROCESSES

2004 ◽  
Vol 14 (06) ◽  
pp. 1987-1993 ◽  
Author(s):  
ANDREAS KAISER ◽  
THOMAS SCHREIBER
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Single Point ◽  
Crucial Point ◽  
Process Dynamics ◽  
Time Index ◽  
Common Time ◽  
Nonparametric Detection ◽  
Definition Of ◽  
Stochastic Point Processes

A new, parameter-free approach based on information theoretical tools is presented which allows the detection of dependences in the dynamics between two point processes. The crucial point is the definition of sequences of inter-event intervals between the events of two stochastic point processes where these sequences are ordered to only one common time index. This is an enhancement of the concept of event intervals of a single point process and makes the analysis of the process dynamics of more than one point processes possible. An application of this method is also illustrated using a model consisting of two synaptically coupled Hindmarsh–Rose neurons.

scholarly journals DISTRIBUTION OF LOCALIZATION CENTERS IN SOME DISCRETE RANDOM SYSTEMS

2007 ◽  
Vol 19 (09) ◽  
pp. 941-965 ◽  
Author(s):  
FUMIHIKO NAKANO
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Anderson Model ◽  
Product Space ◽  
Point Processes ◽  
Random Systems ◽  
Schrödinger Operators ◽  
Scaling Limit ◽  
Random Schrödinger Operators ◽  
Infinitely Divisible

As a supplement of our previous work [10], we consider the localized region of the random Schrödinger operators on l2(Zd) and study the point process composed of their eigenvalues and corresponding localization centers. For the Anderson model we show that, this point process in the natural scaling limit converges in distribution to the Poisson process on the product space of energy and space. In other models with suitable Wegner-type bounds, we can at least show that limiting point processes are infinitely divisible.

Rarefactions of compound point processes

1984 ◽  
Vol 21 (4) ◽  
pp. 710-719 ◽  
Author(s):  
Richard F. Serfozo
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Classical Result ◽  
Point Processes ◽  
Rare Events ◽  
Random Measure ◽  
Bernoulli Trials ◽  
Random Measures ◽  
Infinitely Divisible ◽  
Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

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