Autoregressive approximation in branching processes with a threshold

F. C. Klebaner, O. Nerman

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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.

Original languageEnglish
Pages (from-to)1-7
Number of pages7
JournalStochastic Processes and their Applications
Issue number1
Publication statusPublished - 1 Jan 1994
Externally publishedYes


  • Density dependence
  • Limit cycles
  • Random perturbations

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