Infinitely Divisible Point Processes

1979 ◽  
Vol 30 (5) ◽  
pp. 502
Author(s):  
W. D. Ray ◽  
K. Matthes ◽  
J. Kerstan ◽  
J. Mecke
Keyword(s):  
Point Processes ◽  
Infinitely Divisible

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.

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.

Infinitely Divisible Point Processes.

1980 ◽  
Vol 75 (371) ◽  
pp. 750
Author(s):  
Alan F. Karr ◽  
K. Matthes ◽  
J. Kerstan ◽  
J. Mecke
Keyword(s):  
Point Processes ◽  
Infinitely Divisible

Infinitely Divisible Point Processes

Technometrics ◽  
1982 ◽  
Vol 24 (1) ◽  
pp. 83-83 ◽  
Author(s):  
Peter E. Castro
Keyword(s):  
Point Processes ◽  
Infinitely Divisible

scholarly journals The permanental process

2006 ◽  
Vol 38 (4) ◽  
pp. 873-888 ◽  
Author(s):  
Peter McCullagh ◽  
Jesper Møller
Keyword(s):  
Closed Form ◽  
Large Class ◽  
Point Processes ◽  
Infinitely Divisible ◽  
Determinantal Processes ◽  
Cox Processes

We extend the boson process first to a large class of Cox processes and second to an even larger class of infinitely divisible point processes. Density and moment results are studied in detail. These results are obtained in closed form as weighted permanents, so the extension is called a permanental process. Temporal extensions and a particularly tractable case of the permanental process are also studied. Extensions of the fermion process along similar lines, leading to so-called determinantal processes, are discussed.

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