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Autoregressive approximation in branching processes with a threshold


  • Klebaner, F. C.
  • Nerman, O.


It is shown that population dependent branching processes for large values of threshold can be approximated by Gaussian processes centered at the iterates of the corresponding deterministic function. If the deterministic system has a stable limit cycle, then in the vicinity of the cycle points the corresponding stochastic system can be approximated by an autoregressive process. It is shown that it is possible to speed up convergence to the limit so that the processes converge weakly to the stationary autoregressive process. Similar results hold for noisy dynamical systems when random noise satisfies certain conditions and the corresponding dynamical system has stable limit cycles.

Suggested Citation

  • Klebaner, F. C. & Nerman, O., 1994. "Autoregressive approximation in branching processes with a threshold," Stochastic Processes and their Applications, Elsevier, vol. 51(1), pages 1-7, June.
  • Handle: RePEc:eee:spapps:v:51:y:1994:i:1:p:1-7

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    Cited by:

    1. Klebaner, F. C. & Liptser, R., 1999. "Moderate deviations for randomly perturbed dynamical systems," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 157-176, April.
    2. Ramanan, Kavita & Zeitouni, Ofer, 1999. "The quasi-stationary distribution for small random perturbations of certain one-dimensional maps," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 25-51, November.
    3. Williams, Noah, 2004. "Small noise asymptotics for a stochastic growth model," Journal of Economic Theory, Elsevier, vol. 119(2), pages 271-298, December.
    4. Högnäs, Göran, 1997. "On the quasi-stationary distribution of a stochastic Ricker model," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 243-263, October.


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