Weak convergence of the sum of independent point processes to a Poisson process

1978 ◽  
Vol 17 (1) ◽  
pp. 11-16 ◽  
Author(s):  
R. Banys
Keyword(s):  
Weak Convergence ◽  
Poisson Process ◽  
Point Processes ◽  
Independent Point

Point processes, regular variation and weak convergence

1986 ◽  
Vol 18 (01) ◽  
pp. 66-138 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Weak Convergence ◽  
Function Space ◽  
Regular Variation ◽  
Point Processes ◽  
Sufficient Condition ◽  
Space Setting

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

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.

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.

Densities for stationary random sets and point processes

1984 ◽  
Vol 16 (02) ◽  
pp. 324-346 ◽  
Author(s):  
Wolfgang Weil ◽  
John A. Wieacker
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Random Sets ◽  
Intensity Measure ◽  
Additive Functionals ◽  
Compact Convex Set ◽  
Classical Integral

For certain stationary random setsX, densitiesDφ(X) of additive functionalsφare defined and formulas forare derived whenKis a compact convex set in. In particular, for the quermassintegrals and motioninvariantX, these formulas are in analogy with classical integral geometric formulas. The case whereXis the union set of a Poisson processYof convex particles is considered separately. Here, formulas involving the intensity measure ofYare obtained.

Densities for stationary random sets and point processes

1984 ◽  
Vol 16 (2) ◽  
pp. 324-346 ◽  
Author(s):  
Wolfgang Weil ◽  
John A. Wieacker
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Random Sets ◽  
Intensity Measure ◽  
Additive Functionals ◽  
Compact Convex Set ◽  
Classical Integral

For certain stationary random sets X, densities Dφ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals and motioninvariant X, these formulas are in analogy with classical integral geometric formulas. The case where X is the union set of a Poisson process Y of convex particles is considered separately. Here, formulas involving the intensity measure of Y are obtained.

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.

A martingale approach to the PASTA property

1993 ◽  
Vol 30 (01) ◽  
pp. 252-257
Author(s):  
Michael Scheutzow
Keyword(s):  
Poisson Process ◽  
Queueing Theory ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Boundedness Condition ◽  
Martingale Approach ◽  
Large Numbers ◽  
Poisson Arrivals ◽  
Time Averages ◽  
Image Position

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L 2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.

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.

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