Limit theorems for iterated random functions

Wei Biao Wu, Xiaofeng Shao

Research output: Contribution to journalArticlepeer-review


We study geometric moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under a Dini-continuity condition, a central limit theorem for additive functionals of such systems is established. The empirical processes of sample paths are shown to converge to Gaussian processes in the Skorokhod space. An exponential inequality is established. We present a bound for joint cumulants, which ensures the applicability of several asymptotic results in spectral analysis of time series. Our results provide a vehicle for statistical inferences for fractals and many nonlinear time series models.

Original languageEnglish (US)
Pages (from-to)425-436
Number of pages12
JournalJournal of Applied Probability
Issue number2
StatePublished - Jun 2004
Externally publishedYes


  • Central limit theorem
  • Cumulants
  • Dini continuity
  • Exponential inequality
  • Fractal
  • Iterated random function
  • Markov chain
  • Martingale
  • Nonlinear time series
  • Stationarity

ASJC Scopus subject areas

  • Statistics and Probability
  • General Mathematics
  • Statistics, Probability and Uncertainty


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