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Biometrika 1976 63(1):111-116; doi:10.1093/biomet/63.1.111
© 1976 by Biometrika Trust
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The expected value of the adjusted rescaled Hurst range of independent normal summands

A. A. ANNIS and E. H. LLOYD

Department of Mathematics, Ain Shams University Cairo
Department of Mathematics, Lancaster University

Hurst's empirical law concerning geophysical time series such as annual river flows was framed in terms of an adjusted rescaled range, namely, the range of cumulative sums of deviations of summands from a linearly time-shifted origin, expressed in units of the estimated standard deviation of the full sample. Nonsimulatory theoretical results of the Hurst phenomenon have hitherto been coufined to the adjusted range, without rescaling. The present paper derives the expectation of the adjusted rescaled range for independent normal summands.

Key Words: Geophysical time series • Hurst phenomenon • Random walk • Stochastic hydrology


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