Multidimensional point processes and random closed sets

1984 ◽  
Vol 21 (1) ◽  
pp. 173-178 ◽  
Author(s):  
Michel Baudin
Keyword(s):  
General Result ◽  
Point Processes ◽  
Poisson Processes ◽  
Random Point ◽  
Uniqueness Property ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Mixing Distributions ◽  
Choquet's Theorem ◽  
Special Case

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.

Multidimensional point processes and random closed sets

1984 ◽  
Vol 21 (01) ◽  
pp. 173-178 ◽  
Author(s):  
Michel Baudin
Keyword(s):  
General Result ◽  
Point Processes ◽  
Poisson Processes ◽  
Random Point ◽  
Uniqueness Property ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Mixing Distributions ◽  
Choquet's Theorem ◽  
Special Case

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.

Pair formation in a Markovian arrival process with two event labels

2004 ◽  
Vol 41 (04) ◽  
pp. 1124-1137 ◽  
Author(s):  
Marcel F. Neuts ◽  
Attahiru Sule Alfa
Keyword(s):  
Stochastic Process ◽  
Point Processes ◽  
Pair Formation ◽  
Poisson Processes ◽  
Arrival Process ◽  
Communications Networks ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Interesting Behavior ◽  
Special Case

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

Pair formation in a Markovian arrival process with two event labels

2004 ◽  
Vol 41 (4) ◽  
pp. 1124-1137 ◽  
Author(s):  
Marcel F. Neuts ◽  
Attahiru Sule Alfa
Keyword(s):  
Stochastic Process ◽  
Point Processes ◽  
Pair Formation ◽  
Poisson Processes ◽  
Arrival Process ◽  
Communications Networks ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Interesting Behavior ◽  
Special Case

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

On doubly stochastic Poisson processes

1964 ◽  
Vol 60 (4) ◽  
pp. 923-930 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Poisson Process ◽  
Stationary Point ◽  
Point Processes ◽  
Poisson Processes ◽  
Doubly Stochastic ◽  
Doubly Stochastic Poisson Process ◽  
Stationary Point Process ◽  
Special Case ◽  
Stationary Point Processes ◽  
Made In

The class of stationary point processes known as ‘doubly stochastic Poisson processes’ was introduced by Cox (2) and has been studied in detail by Bartlett (1). It is not clear just how large this class is, and indeed it seems to be a problem of some difficulty to decide of a general stationary point process whether or not it can be represented as a doubly stochastic Poisson process. (A few simple necessary conditions are known. For instance, Cox pointed out in the discussion to (1) that a double stochastic Poisson process must show more ‘dispersion’ than the Poisson process. Such conditions are very far from being sufficient.) The main result of the present paper is a solution of the problem for the special case of a renewal process, justifying an assertion made in the discussion to (1).

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.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (4) ◽  
pp. 881-889 ◽  
Author(s):  
Hans Dieter Unkelbach
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Road Traffic ◽  
Poisson Processes ◽  
Traffic Model ◽  
Cluster Point ◽  
Convergence To Equilibrium ◽  
Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

Random point processes

1977 ◽  
Vol 65 (6) ◽  
pp. 990-990
Author(s):  
I. Rubin
Keyword(s):  
Point Processes ◽  
Random Point
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