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Point processes with finite-dimensional conditional probabilities

Author

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  • Asmussen, Søren
  • Bladt, Mogens

Abstract

We study the structure of point processes N with the property that the vary in a finite-dimensional space where [theta]t is the shift and the [sigma]-field generated by the counting process up to time t. This class of point processes is strictly larger than Neuts' class of Markovian arrival processes. On the one hand, it allows for more general features like interarrival distributions which are matrix-exponential rather than phase type, on the other the probabilistic interpretation is a priori less clear. Nevertheless, the properties are very similar. In particular, finite-dimensional distributions of interarrival times, moments, Laplace transforms, Palm distributions, etc., are shown to be given by two fundamental matrices C, D just as for the Markovian arrival process. We also give a probabilistic interpretation in terms of a piecewise deterministic Markov process on a compact convex subset of , whose jump times are identical to the epochs of N.

Suggested Citation

  • Asmussen, Søren & Bladt, Mogens, 1999. "Point processes with finite-dimensional conditional probabilities," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 127-142, July.
    • Handle: RePEc:eee:spapps:v:82:y:1999:i:1:p:127-142
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    Citations

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    Cited by:

    1. Dheeraj Goyal & Nil Kamal Hazra & Maxim Finkelstein, 2022. "On Properties of the Phase-type Mixed Poisson Process and its Applications to Reliability Shock Modeling," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2933-2960, December.
    2. Søren Asmussen & Mogens Bladt, 2022. "From PH/MAP to ME/RAP," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 173-175, April.
    3. Bodrog, L. & Heindl, A. & Horváth, G. & Telek, M., 2008. "A Markovian canonical form of second-order matrix-exponential processes," European Journal of Operational Research, Elsevier, vol. 190(2), pages 459-477, October.
    4. Bean, Nigel G. & Nguyen, Giang T. & Nielsen, Bo F. & Peralta, Oscar, 2022. "RAP-modulated fluid processes: First passages and the stationary distribution," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 308-340.
    5. Levente Bodrog & András Horváth & Miklós Telek, 2008. "Moment characterization of matrix exponential and Markovian arrival processes," Annals of Operations Research, Springer, vol. 160(1), pages 51-68, April.
    6. Miklós Telek, 2022. "The two-matrix problem," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 265-267, April.
    7. Peter Buchholz & Miklós Telek, 2013. "Rational Automata Networks: A Non-Markovian Modeling Approach," INFORMS Journal on Computing, INFORMS, vol. 25(1), pages 87-101, February.
    8. Peter Buchholz & Miklós Telek, 2013. "On minimal representations of Rational Arrival Processes," Annals of Operations Research, Springer, vol. 202(1), pages 35-58, January.

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