Ergodic Properties of Infinitely Divisible Stochastic Point Processes

1981 ◽  
Vol 102 (1) ◽  
pp. 127-135 ◽  
Author(s):  
Klaus Fleischmann
Keyword(s):  
Point Processes ◽  
Infinitely Divisible ◽  
Ergodic Properties ◽  
Stochastic 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.

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.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

On Updating Algorithms and Inference for Stochastic Point Processes

1975 ◽  
Vol 12 (S1) ◽  
pp. 239-259 ◽  
Author(s):  
D. Vere-Jones
Keyword(s):  
Stochastic Process ◽  
Maximum Likelihood ◽  
Stochastic Approximation ◽  
Approximation Theory ◽  
Point Processes ◽  
Maximum Likelihood Estimates ◽  
Doubly Stochastic ◽  
Stochastic Point Processes ◽  
Asymptotically Efficient ◽  
Estimation And Filtering

This paper is an attempt to interpret and extend, in a more statistical setting, techniques developed by D. L. Snyder and others for estimation and filtering for doubly stochastic point processes. The approach is similar to the Kalman-Bucy approach in that the updating algorithms can be derived from a Bayesian argument, and lead ultimately to equations which are similar to those occurring in stochastic approximation theory. In this paper the estimates are derived from a general updating formula valid for any point process. It is shown that almost identical formulae arise from updating the maximum likelihood estimates, and on this basis it is suggested that in practical situations the sequence of estimates will be consistent and asymptotically efficient. Specific algorithms are derived for estimating the parameters in a doubly stochastic process in which the rate alternates between two levels.

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.

A stochastic process whose successive intervals between events form a first order Markov chain — I

1968 ◽  
Vol 5 (03) ◽  
pp. 648-668
Author(s):  
D. G. Lampard
Keyword(s):  
Markov Chain ◽  
Point Processes ◽  
Electronic Device ◽  
Statistical Properties ◽  
Time Intervals ◽  
First Order ◽  
Stochastic Point Process ◽  
Order Markov Chain ◽  
The One ◽  
Stochastic Point Processes

In this paper we discuss a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain. Such processes may be regarded as next, in order of complexity, in a hierarchy of stochastic point processes, to “renewal” processes, which latter have been studied extensively. The main virtue of the particular system which is studied here is that virtually all its important statistical properties can be obtained in closed form and that it is physically realizable as an electronic device. As such it forms the basis for a laboratory generator whose output may be used for experimental work involving processes of this kind. Such statistical properties as the one and two-dimensional probability densities for the time intervals are considered in both the stationary and nonstationary state and also discussed are corresponding properties of the successive numbers arising in the stores of the counter system. In particular it is shown that the degree of coupling between successive time intervals may be adjusted in practice without altering the one dimensional probability density for the interval lengths. It is pointed out that operation of the counter system may also be regarded as a problem in queueing theory involving one server alternately serving two queues. A generalization of the counter system, whose inputs are normally a pair of statistically independent Poisson processes, to the case where one of the inputs is a renewal process is considered and leads to some interesting functional equations.

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.

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