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The Hurst effect under trends

Published online by Cambridge University Press:  14 July 2016

R. N. Bhattacharya ,
Vijay K. Gupta  and
Ed Waymire
R. N. Bhattacharya*
Affiliation:
Indiana University
Vijay K. Gupta*
Affiliation:
University of Mississippi
Ed Waymire*
Affiliation:
Oregon State University
*
∗ Postal address: Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.
∗∗ Postal address: Department of Civil Engineering, University of Mississippi, University, MS 38677, U.S.A.
∗∗∗ Postal address: Department of Mathematics, Oregon State University, Corvallis, OR 97331, U.S.A.
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Abstract

Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the form f(n) = c(m + n)ß, where m is an arbitrary non-negative parameter and c is not 0. For – ½ < ß < 0 the Hurst exponent is shown to be precisely given by 1 + ß. For ß ≦ – ½ and for ß = 0 the Hurst exponent is 0.5, while for ß > 0 it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.

Keywords

HURST EXPONENTWEAKLY DEPENDENT RANDOM VARIABLESGEOPHYSICAL MODELS
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 3 , September 1983 , pp. 649 - 662
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

This research was partially supported by Grants CME-7907793 and MCS-8201628 from the National Science Foundation.

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