A Markov construction for a multidimensional point process

1977 ◽  
Vol 14 (3) ◽  
pp. 507-515 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Average Density ◽  
Second Order ◽  
Finite Point ◽  
Markov Sequence

A sequence of finite point processes {Pn} is constructed in using a Markov sequence of points. Essentially, in the process Pn consisting of n events, the coordinates of these events are simply the first n points of a Markov sequence suitably scaled so that the average density of the process is independent of n. The second-order properties of Pn are discussed and sufficient conditions are found for Pn to converge in distribution to a Poisson process as n →∞. A simple example involving the cardioid distribution is described.

A Markov construction for a multidimensional point process

1977 ◽  
Vol 14 (03) ◽  
pp. 507-515 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Average Density ◽  
Second Order ◽  
Finite Point ◽  
Markov Sequence

A sequence of finite point processes {Pn } is constructed in using a Markov sequence of points. Essentially, in the process Pn consisting of n events, the coordinates of these events are simply the first n points of a Markov sequence suitably scaled so that the average density of the process is independent of n. The second-order properties of Pn are discussed and sufficient conditions are found for Pn to converge in distribution to a Poisson process as n →∞. A simple example involving the cardioid distribution is described.

Dependent thinning of point processes

1980 ◽  
Vol 17 (04) ◽  
pp. 987-995 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Compound Poisson Process ◽  
Second Order ◽  
Binary Variables ◽  
Compound Poisson ◽  
The Real ◽  
Markov Sequence

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.

Dependent thinning of point processes

1980 ◽  
Vol 17 (4) ◽  
pp. 987-995 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Compound Poisson Process ◽  
Second Order ◽  
Binary Variables ◽  
Compound Poisson ◽  
The Real ◽  
Markov Sequence

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.

scholarly journals Poisson convergence for point processes on the plane

1985 ◽  
Vol 39 (2) ◽  
pp. 253-269 ◽  
Author(s):  
B. Gail Ivanoff
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Convergence In Distribution ◽  
Deterministic Function ◽  
Skorokhod Topology

AbstractA compensator is defined for a point process in two dimensions. It is shown that a Poisson process is characterized by a continuous deterministic compensator. Sufficient conditions are given for convergence in distribution of a sequence of two-dimensional point processes in the Skorokhod topology to a Poisson process when the corresponding sequence of compensators converges pointwise in probability to a continuous deterministic function.

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.

A limit theorem for spatial point processes

1986 ◽  
Vol 18 (03) ◽  
pp. 646-659 ◽  
Author(s):  
Steven P. Ellis
Keyword(s):  
Limit Theorem ◽  
Poisson Process ◽  
Asymptotic Distribution ◽  
Point Process ◽  
Point Processes ◽  
Affine Transformations ◽  
Density Estimates ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
As If

Spatial point processes are considered whose points are subjected to certain classes of affine transformations indexed by some variable, T. Under some hypotheses, for large T integrals with respect to such a point process behave approximately as if the process were Poisson. Under stronger hypotheses, the transformed process converges as a process to a Poisson process. The result gives the asymptotic distribution of certain density estimates.

State aggregation and discrete-state Markov chains embedded in a class of point processes

1995 ◽  
Vol 32 (01) ◽  
pp. 39-51
Author(s):  
Xi-Ren Cao
Keyword(s):  
Markov Chains ◽  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
Practical Importance ◽  
Marked Point Process ◽  
Interarrival Time ◽  
Discrete State ◽  
State Aggregation

One result that is of both theoretical and practical importance regarding point processes is the method of thinning. The basic idea of this method is that under some conditions, there exists an embedded Poisson process in any point process such that all its arrival points form a sub-sequence of the Poisson process. We extend this result by showing that on the embedded Poisson process of a uni- or multi-variable marked point process in which interarrival time distributions may depend on the marks, one can define a Markov chain with a discrete state that characterizes the stage of the interarrival times. This implies that one can construct embedded Markov chains with countable state spaces for the state processes of many practical systems that can be modeled by such point processes.

scholarly journals Poisson approximation for some point processes in reliability

2004 ◽  
Vol 36 (2) ◽  
pp. 455-470 ◽  
Author(s):  
Jean-Bernard Gravereaux ◽  
James Ledoux
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Poisson Approximation ◽  
Arrival Process ◽  
Homogeneous Poisson Process ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Finite Markov Process ◽  
Failure Point

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

scholarly journals Poisson approximation for some point processes in reliability

2004 ◽  
Vol 36 (02) ◽  
pp. 455-470
Author(s):  
Jean-Bernard Gravereaux ◽  
James Ledoux
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Poisson Approximation ◽  
Arrival Process ◽  
Homogeneous Poisson Process ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Finite Markov Process ◽  
Failure Point

In this paper, we consider a failure point process related to the Markovian arrival process defined by Neuts. We show that it converges in distribution to a homogeneous Poisson process. This convergence takes place in the context of rare occurrences of failures. We also provide a convergence rate of the convergence in total variation of this point process using an approach developed by Kabanov, Liptser and Shiryaev for the doubly stochastic Poisson process driven by a finite Markov process.

scholarly journals Poisson limits for pairwise and area interaction point processes

2000 ◽  
Vol 32 (1) ◽  
pp. 75-85 ◽  
Author(s):  
S. Rao Jammalamadaka ◽  
Mathew D. Penrose
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Compact Support ◽  
Point Processes ◽  
Joint Density ◽  
Interaction Point

Suppose n particles xi in a region of the plane (possibly representing biological individuals such as trees or smaller organisms) have a joint density proportional to exp{-∑i<jϕ(n(xi-xj))}, with ℝd; a specified function of compact support. We obtain a Poisson process limit for the collection of rescaled interparticle distances as n becomes large. We give corresponding results for the case of several types of particles, representing different species, and also for the area-interaction (Widom-Rowlinson) point process of interpenetrating spheres.

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