ON THE STRENGTH OF DEPENDENCE OF A TIME SERIES GENERATED BY A CHAOTIC MAP

Peter Hall and Rodney C. L. Wolff

Journal of Time Series Analysis, 1995, vol. 16, issue 6, 571-583

Abstract: Abstract. A stochastic sequence generated by a chaotic map has extremely strong dependence in a structural sense, in that any data value may be represented exactly as a known deterministic function of any one of its antecedents. However, the range of dependence of the time series may be very short in a statistical sense ‐ in fact, all its lagged correlations could be zero. In the present paper we study the implications of this property for two of the statistical techniques which weak dependence is often invoked to justify ‐ asymptotic methods based on the central limit theorem, and the bootstrap. It is shown that in the case of the logistic map, the validity of these techniques depends critically on the value of the parameter governing the map. Very small alterations to the parameter value can produce dramatic changes in the strength of dependence, thereby altering the validity of even elementary statistical procedures based on asymptotic normality or resampling.

Date: 1995
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