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A self-correcting point process

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  • Isham, Valerie
  • Westcott, Mark

Abstract

Suppose a point process is attempting to operate as closely as possible to a deterministic rate [rho], in the sense of aiming to produce [rho]t points during the interval (0,t] for all t. This can be modelled by making the instantaneous rate of t of the process a suitable function of n-[rho]t, n being the number of points in [0, t]. This paper studies such a self-correcting point process in two cases: when the point process is Markovian and the rate function is very general, and when the point process is arbitrary and the rate function is exponential. In each case it is shown that as t-->[infinity] the mean number of points occuring in (0, t] is [rho]t+O(1) while the variance is bounded further, in the Markov case all the absolute central moments are bounded. An application to the outputs of stationary D/M/s queues is given.

Suggested Citation

  • Isham, Valerie & Westcott, Mark, 1979. "A self-correcting point process," Stochastic Processes and their Applications, Elsevier, vol. 8(3), pages 335-347, May.
    • Handle: RePEc:eee:spapps:v:8:y:1979:i:3:p:335-347
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    Citations

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    Cited by:

    1. Zhu, Lingjiong, 2013. "Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 544-550.
    2. Jakob G. Rasmussen & Heidi S. Christensen, 2021. "Point Processes on Directed Linear Networks," Methodology and Computing in Applied Probability, Springer, vol. 23(2), pages 647-667, June.
    3. Frederic Schoenberg, 2002. "On Rescaled Poisson Processes and the Brownian Bridge," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(2), pages 445-457, June.
    4. Scalas, Enrico & Kaizoji, Taisei & Kirchler, Michael & Huber, Jürgen & Tedeschi, Alessandra, 2006. "Waiting times between orders and trades in double-auction markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 463-471.
    5. Philip A. White & Alan E. Gelfand, 2021. "Generalized Evolutionary Point Processes: Model Specifications and Model Comparison," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 1001-1021, September.
    6. Peter Halpin & Paul Boeck, 2013. "Modelling Dyadic Interaction with Hawkes Processes," Psychometrika, Springer;The Psychometric Society, vol. 78(4), pages 793-814, October.
    7. Nobuo Inagaki & Toshihabu Hayashi, 1990. "Parameter estimation for the simple self-correcting point process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(1), pages 89-98, March.

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