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Limit theorems for iterated random functions

Part of: Stochastic processes Markov processes Limit theorems

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.
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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 

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