scholarly journals Nearest-neighbour Markov point processes on graphs with Euclidean edges

2018 ◽  
Vol 50 (4) ◽  
pp. 1275-1293 ◽  
Author(s):  
M. N. M. van Lieshout
Keyword(s):  
Real Line ◽  
Point Processes ◽  
Local Geometry ◽  
Renewal Processes ◽  
Nearest Neighbour ◽  
The Real ◽  
Consistency Conditions ◽  
Linear Networks ◽  
Markov Point Processes ◽  
Neighbour Point

Abstract We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles.

scholarly journals The probability generating functional

1972 ◽  
Vol 14 (4) ◽  
pp. 448-466 ◽  
Author(s):  
M. Westcott
Keyword(s):  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Generating Functional ◽  
The Real ◽  
The Subject

This paper is concerned with certain aspects of the theory and application of the probability generating functional (p.g.fl) of a point process on the real line. Interest in point processes has increased rapidly during the last decade and a number of different approaches to the subject have been expounded (see for example [6], [11], [15], [17], [20], [25], [27], [28]). It is hoped that the present development using the p.g.ff will calrify and unite some of these viewpoints and provide a useful tool for solution of particular problems.

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.

Sufficient conditions for long-range count dependence of stationary point processes on the real line

2001 ◽  
Vol 38 (2) ◽  
pp. 570-581 ◽  
Author(s):  
Rafał Kulik ◽  
Ryszard Szekli
Keyword(s):  
Long Range ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Second Moment ◽  
The Real ◽  
Palm Measure ◽  
Stationary Point Processes

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

A spatial Markov property for nearest-neighbour Markov point processes

1990 ◽  
Vol 27 (04) ◽  
pp. 767-778 ◽  
Author(s):  
W. S. Kendall
Keyword(s):  
Point Processes ◽  
Markov Property ◽  
Nearest Neighbour ◽  
Markov Point Processes

Nearest-neighbour Markov point processes were introduced by Baddeley and Møller (1989) as generalizations of the Markov point processes of Ripley and Kelly. This note formulates and discusses a spatial Markov property for these point processes.

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 On the weak convergence of stochastic processes with embedded point processes

1988 ◽  
Vol 20 (2) ◽  
pp. 473-475 ◽  
Author(s):  
Panagiotis Konstantopoulos ◽  
Jean Walrand
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Point Processes ◽  
Mixing Condition ◽  
The Real ◽  
Palm Distribution

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

A versatile Markovian point process

1979 ◽  
Vol 16 (4) ◽  
pp. 764-779 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Point Processes ◽  
Probability Distributions ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Probability Models ◽  
Phase Type ◽  
Finite State ◽  
Markov Point Processes ◽  
Arrival Processes ◽  
Markov Modulated

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

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