An Introduction to the Theory of Point Processes

Daley, D. J.; Vere-Jones, D.

doi:10.1007/978-1-4757-2001-3

Cited by 1,579 publications

(3,034 citation statements)

References 0 publications

Supporting

Mentioning

2,983

Contrasting

Unclassified

Order By: Relevance

“…The fundamentals of the theory of marked point processes can be found in Daley & Vere-Jones (2004) and Stoyan et al (1995). Here only the facts needed in the present paper are given.…”

Section: Summary Characteristics For Marked Point Processesmentioning

confidence: 99%

Modelling marked point patterns by intensity-marked Cox processes

Stoyan

2008

Statistics & Probability Letters

View full text Add to dashboard Cite

show abstract

“…The fundamentals of the theory of marked point processes can be found in Daley & Vere-Jones (2004) and Stoyan et al (1995). Here only the facts needed in the present paper are given.…”

Section: Summary Characteristics For Marked Point Processesmentioning

confidence: 99%

Modelling marked point patterns by intensity-marked Cox processes

Stoyan

2008

Statistics & Probability Letters

View full text Add to dashboard Cite

show abstract

“…If (60) holds under the Palm probability P 0 of the stationary sequence Z n and T 1 is under P 0 independent of Z 1 Z 2 , then (60) also holds underP . This class includes marks independent of the point process, the stationary ON/OFF process, and, more generally, any MPP with unpredictable marks; see Definition 6.4.III in Daley and Vere-Jones [7].…”

Section: Proposition 52 Assume the Following Conditions Hold: (I) Tmentioning

confidence: 99%

Scaling Limits for Cumulative Input Processes

Mikosch

Samorodnitsky

2007

Mathematics of OR

View full text Add to dashboard Cite

We study different scaling behavior of very general telecommunications cumulative input processes. The activities of a telecommunication system are described by a marked-point process T n Z n n∈ , where T n is the arrival time of a packet brought to the system or the starting time of the activity of an individual source, and the mark Z n is the amount of work brought to the system at time T n . This model includes the popular ON/OFF process and the infinite-source Poisson model. In addition to the latter models, one can flexibly model dependence of the interarrival times T n − T n−1 , clustering behavior due to the arrival of an impulse generating a flow of activities, but also dependence between the arrival process T n and the marks Z n . Similarly to the ON/OFF and infinite-source Poisson model, we can derive a multitude of scaling limits for the input process of one source or for the superposition of an increasing number of such sources. The memory in the input process depends on a variety of factors, such as the tails of the interarrival times or the tails of the distribution of activities initiated at an arrival T n , or the number of activities starting at T n . It turns out that, as in standard results on the scaling behavior of cumulative input processes in telecommunications, fractional Brownian motion or infinite-variance Lévy stable motion can occur in the scaling limit. However, the fractional Brownian motion is a much more robust limit than the stable motion, and many other limits may occur as well.

show abstract

“…Let N ≡ {N (A)} A∈E be a point process defined on a probability space ( , F , P), whose realizations are integer-valued, boundedly finite measures on (E, E ); see [3,Section 7.1]. Given A ∈ E , by F A we will denote the sub-σ -field of F generated by the random variables {N(B) : B ∈ E , B ⊆ A}.…”

Section: Basic Definitionsmentioning

confidence: 99%

“…From Theorem 7.1.XI of [3], it is straightforward to derive necessary and sufficient conditions ensuring that a family of probability vectors p A and lower triangular stochastic matrices P A,B determines the distribution of a Markovian point process. …”

Section: Markovian Point Processesmentioning

confidence: 99%

Conditionally Independent Increment Point Processes

Ibarrola

Prieto-Rumeau

2011

J. Appl. Probab.

View full text Add to dashboard Cite

In this paper we introduce conditionally independent increment point processes, that is, processes that are conditionally independent inside and outside a bounded set A given N(A), the number of points in A. We show that these point processes can be characterized by means of the avoidance function of a multinomial 'support process', the solution of a suitably defined linear system of equations, and, finally, the infinitesimal matrix of a continuous-time Markov chain.

show abstract

An Introduction to the Theory of Point Processes

Abstract: Library of Congress Cataloging-in-Publication Data Daley, Daryl J.An introduction to the theory of point processes.(Springer series in statistics) Includes bibliographies. 1. Point processes. 1. Vere-Jones, D.

Cited by 1,579 publications

References 0 publications

Modelling marked point patterns by intensity-marked Cox processes

Modelling marked point patterns by intensity-marked Cox processes

Scaling Limits for Cumulative Input Processes

Conditionally Independent Increment Point Processes

Contact Info

Product

Resources

About