Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-20T00:13:42.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for iterated random functions

Part of: Markov processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Wei Biao Wu  and
Xiaofeng Shao
Wei Biao Wu*
Affiliation:
University of Chicago
Xiaofeng Shao*
Affiliation:
University of Chicago
*
∗ Postal address: Department of Statistics, University of Chicago, Chicago, IL 60637, USA.
∗ Postal address: Department of Statistics, University of Chicago, Chicago, IL 60637, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Stationarityiterated random functioncentral limit theoremDini continuityexponential inequalitymartingaleMarkov chainfractalnonlinear time seriescumulants

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G42: Martingales with discrete parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 425 - 436
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alsmeyer, G., and Fuh, C. D. (2001). Limit theorems for iterated random functions by regenerative methods. Stoch. Process. Appl. 96, 123–142.Google Scholar
Anderson, T. W. (1971). The Statistical Analysis of Time Series. John Wiley, New York.Google Scholar
Arnold, L. (1998). Random Dynamical Systems. Springer, Berlin.Google Scholar
Barnsley, M. F., and Elton, J. H. (1988). A new class of Markov processes for image encoding. Adv. Appl. Prob. 20, 14–32 Google Scholar
Bosq, D. (1996). Nonparametric Statistics for Stochastic Processes. Estimation and Prediction (Lecture Notes Statist. 110). Springer, New York.Google Scholar
Brillinger, D. R. (1981). Time Series: Data Analysis and Theory, 2nd edn. Holden-Day, San Francisco.Google Scholar
Chan, K. S., and Tong, H. (2001). Chaos: A Statistical Perspective. Springer, New York.Google Scholar
Diaconis, P. and Freedman., D. (1999). Iterated random functions. SIAM Rev. 41, 41–76.Google Scholar
Doukhan, P., and Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stoch. Process. Appl. 84, 313–342.Google Scholar
Elton, J. H. (1990). A multiplicative ergodic theorem for Lipschitz maps. Stoch. Process. Appl. 34, 39–47.Google Scholar
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.Google Scholar
Freedman, D. A. (1975). On tail probabilities for martingales. Ann. Prob. 3, 100–118.Google Scholar
Gordin, M. I. and Lifšic, B. A. (1978). Central limit theorem for stationary Markov processes. Dokl. Adad. Nauk SSSR 239, 766–767 (in Russian). English translation: Soviet Math. Dokl. 19 (1978), 392–394.Google Scholar
Herkenrath, U., Iosifescu, M., and Rudolph, A. (2003). A note on invariance principles for iterated random functions. J. Appl. Prob. 40, 834–837.Google Scholar
Hutchinson, J. (1981). Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747.Google Scholar
Jarner, S. F., and Tweedie, R.L. (2001). Locally contracting iterated functions and stability of Markov chains. J. Appl. Prob. 38, 494–507.CrossRefGoogle Scholar
Nicholls, D. F., and Quinn, B. G. (1982). Random Coefficient Autoregressive Models: An Introduction. Springer, New York.CrossRefGoogle Scholar
Petruccelli, J. D., and Woolford, S. W. (1984). A threshold AR(1) model. J. Appl. Prob. 21, 270–286 CrossRefGoogle Scholar
Rosenblatt, M. (1984). Asymptotic normality, strong mixing, and spectral density estimates. Ann. Prob. 12, 1167–1180.Google Scholar
Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston, MA.Google Scholar
Solomyak, B. M. (1995). On the random series Σ ±λ n (an Erdõs problem). Ann. Math. 142, 611–625.Google Scholar
Steinsaltz, D. (1999). Locally contractive iterated function systems. Ann. Prob. 27, 1952–1979.Google Scholar
Stenflo, Ö. (1998). Ergodic theorems for iterated function systems controlled by stochastic sequences. Doctoral Thesis No. 14, Umeå University.Google Scholar
Tong, H. (1990). Nonlinear Time Series. A Dynamical System Approach. Oxford University Press.Google Scholar
Wu, W. B., and Woodroofe, M. (2000). A central limit theorem for iterated random functions. J. Appl. Prob. 37, 748–755.Google Scholar