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The Hurst effect under trends

  • R. N. Bhattacharya (a1), Vijay K. Gupta (a2) and Ed Waymire (a3)


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.


Corresponding author

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



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