Count distributions, orderliness and invariance of Poisson cluster processes

1979 ◽  
Vol 16 (02) ◽  
pp. 261-273 ◽  
Author(s):  
Larry P. Ammann ◽  
Peter F. Thall
Keyword(s):  
Present Article ◽  
Generating Functional ◽  
Cluster Process ◽  
Multiple Events ◽  
Poisson Cluster Process ◽  
Finite Dimensional ◽  
The Family ◽  
Moment Measures ◽  
Poisson Cluster ◽  
Cluster Processes

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.

Count distributions, orderliness and invariance of Poisson cluster processes

1979 ◽  
Vol 16 (2) ◽  
pp. 261-273 ◽  
Author(s):  
Larry P. Ammann ◽  
Peter F. Thall
Keyword(s):  
Present Article ◽  
Generating Functional ◽  
Cluster Process ◽  
Multiple Events ◽  
Poisson Cluster Process ◽  
Finite Dimensional ◽  
The Family ◽  
Moment Measures ◽  
Poisson Cluster ◽  
Cluster Processes

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.

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.

Note on the determination of cluster centers from a realization of a multidimensional Poisson cluster process

1983 ◽  
Vol 20 (01) ◽  
pp. 136-143
Author(s):  
Michel Baudin
Keyword(s):  
Point Processes ◽  
Linear Method ◽  
Joint Probability ◽  
Observation Window ◽  
Generating Functional ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
Actual Use ◽  
Poisson Cluster

This is the sequel to a previous paper (Baudin (1981)). The joint probability generating functional of two point processes is introduced as a tool to compute the conditional intensity of the process of cluster centers of a multidimensional Poisson cluster process when a realization is given in a bounded observation window. An explicit formula is derived but it is too complicated for actual use; a linear method for practical estimation is discussed.

Some limit theorems for clustered occupancy models

1983 ◽  
Vol 20 (04) ◽  
pp. 788-802
Author(s):  
Larry P. Ammann
Keyword(s):  
Present Article ◽  
Limit Theorems ◽  
Poisson Processes ◽  
Occupancy Models ◽  
Limiting Distributions ◽  
Occupancy Model ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
The Individual ◽  
Poisson Cluster

Most generalizations of the classical occupancy model involve non-homogeneous shot assignment probabilities, but retain the independence of the individual shot assignments. Hence, these models are associated with non-homogeneous Poisson processes. The present article discusses a generalization in which the shot assignments are not independent, but which result in clustering of the shots. Conditions are given under which this clustered occupancy model converges to a Poisson cluster process. Limiting distributions for the number of empty cells are obtained for various allocation intensities when the total number of shots is deterministic as well as random. In particular, it is shown that when the allocation is sparse, then the limiting distribution of the number of empty cells is compound Poisson.

Note on the determination of cluster centers from a realization of a multidimensional Poisson cluster process

1983 ◽  
Vol 20 (1) ◽  
pp. 136-143 ◽  
Author(s):  
Michel Baudin
Keyword(s):  
Point Processes ◽  
Linear Method ◽  
Joint Probability ◽  
Observation Window ◽  
Generating Functional ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
Actual Use ◽  
Poisson Cluster

This is the sequel to a previous paper (Baudin (1981)). The joint probability generating functional of two point processes is introduced as a tool to compute the conditional intensity of the process of cluster centers of a multidimensional Poisson cluster process when a realization is given in a bounded observation window. An explicit formula is derived but it is too complicated for actual use; a linear method for practical estimation is discussed.

A cluster process representation of a self-exciting process

1974 ◽  
Vol 11 (03) ◽  
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.

Some limit theorems for clustered occupancy models

1983 ◽  
Vol 20 (4) ◽  
pp. 788-802 ◽  
Author(s):  
Larry P. Ammann
Keyword(s):  
Present Article ◽  
Limit Theorems ◽  
Poisson Processes ◽  
Occupancy Models ◽  
Limiting Distributions ◽  
Occupancy Model ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
The Individual ◽  
Poisson Cluster

Most generalizations of the classical occupancy model involve non-homogeneous shot assignment probabilities, but retain the independence of the individual shot assignments. Hence, these models are associated with non-homogeneous Poisson processes. The present article discusses a generalization in which the shot assignments are not independent, but which result in clustering of the shots. Conditions are given under which this clustered occupancy model converges to a Poisson cluster process. Limiting distributions for the number of empty cells are obtained for various allocation intensities when the total number of shots is deterministic as well as random. In particular, it is shown that when the allocation is sparse, then the limiting distribution of the number of empty cells is compound Poisson.

Results in the asymptotic and equilibrium theory of Poisson cluster processes

1973 ◽  
Vol 10 (4) ◽  
pp. 807-823 ◽  
Author(s):  
M. Westcott
Keyword(s):  
Asymptotic Distribution ◽  
General Formula ◽  
Real Line ◽  
Probability Generating Function ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
Asymptotic Formulae ◽  
Equilibrium Process ◽  
Poisson Cluster ◽  
Cluster Processes

This paper contains a detailed study of the Poisson cluster process on the real line, concentrating on two aspects; first, the asymptotic distribution of the number of points in [0,t) as t→ ∞ for both transient and equilibrium cluster processes and, secondly, a general formula for the probability generating function of the equilibrium process. Asymptotic formulae for cumulants of the process are also derived. The results obtained generalize those of previous writers. The approach is analytical, in contrast to the probabilistic treatment of P. A. W. Lewis.

Results in the asymptotic and equilibrium theory of Poisson cluster processes

1973 ◽  
Vol 10 (04) ◽  
pp. 807-823 ◽  
Author(s):  
M. Westcott
Keyword(s):  
Asymptotic Distribution ◽  
General Formula ◽  
Real Line ◽  
Probability Generating Function ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
Asymptotic Formulae ◽  
Equilibrium Process ◽  
Poisson Cluster ◽  
Cluster Processes

This paper contains a detailed study of the Poisson cluster process on the real line, concentrating on two aspects; first, the asymptotic distribution of the number of points in [0,t) as t→ ∞ for both transient and equilibrium cluster processes and, secondly, a general formula for the probability generating function of the equilibrium process. Asymptotic formulae for cumulants of the process are also derived. The results obtained generalize those of previous writers. The approach is analytical, in contrast to the probabilistic treatment of P. A. W. Lewis.

Cluster shock models

1981 ◽  
Vol 18 (01) ◽  
pp. 104-111 ◽  
Author(s):  
Peter F. Thall
Keyword(s):  
Failure Rate ◽  
Present Note ◽  
Cluster Process ◽  
Completely Monotonic ◽  
Poisson Cluster Process ◽  
Preservation Theorem ◽  
Shock Models ◽  
Poisson Cluster ◽  
Decreasing Failure Rate ◽  
Over Time

The survival distribution of a device subject to a sequence of shocks occurring randomly over time is studied by Esary, Marshall and Proschan (1973) and by A-Hameed and Proschan (1973), (1975). The present note treats the case in which shocks occur according to a homogeneous Poisson cluster process. It is shown that if[the device surviveskshocks] =zk, 0 <z< 1, then the device exhibits a decreasing failure rate. A DFR preservation theorem is proved for completely monotonic. A counterexample to the IFR preservation theorem is given in whichis strictly IFR while the failure rate is initially decreasing and then increasing.

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