Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T08:55:04.905Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

Boes, D. C. and Salas-Lacruz, J. D. (1978) Nonstationarity of the mean and the Hurst phenomenon. Water Resources Res. 14, 135–143.CrossRefGoogle Scholar
Boes, D. C. and Salas-Lacruz, J. D. (1973) On the expected range of partial sums of exchangeable random variables J. Appl. Prob. 10, 671–677.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Burton, R. and Waymire, E. (1983) Limit theorems for point random fields. J. Math. Anal. Google Scholar
Doob, J. L. (1949) Heuristic approach to the Kolmogorov-Smirnov theorem. Ann. Math. Statist. 20, 393–403.CrossRefGoogle Scholar
Feller, W. (1951) The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Statist. 22, 427–432.Google Scholar
Gordin, M. I. and Lifšic, B. A. (1978) The central limit theorem for stationary Markov processes. Dokl. Akad. Nauk SSSR 19, 392–394.Google Scholar
Hurst, H. E. (1951) Long term storage capacity of reservoirs. Trans. Amer. Soc. Civil Engrs. 116, 770–779.Google Scholar
Hurst, H. E., Black, R. P. and Simaika, Y. M. (1965) Long-Term Storage: An Experimental Study. Constable Press, London.Google Scholar
Ibragimov, I. A. and Linnik, Y. V. (1971) Independent and Stationary Sequences of Random Variables (English translation). Wolters-Noordhoff, Gronigen.Google Scholar
Klemes, V. (1974) The Hurst phenomenon — a puzzle? Water Resources Res. 10, 675–688.Google Scholar
Klemes, V. (1978) Physically based stochastic hydrologic analysis. Adv. Hydroscience 11, 285–356.CrossRefGoogle Scholar
Klemes, V. (1982) Empirical and causal models in hydrology. In Studies in Geophysics, National Academy Press, Washington, D.C., 95–104.Google Scholar
Kottegoda, N. T. (1970) Applicability of short-memory model to English river flow data. J. Inst. Water Engrs. 24, 481–489.Google Scholar
Leopold, L. B., Wolman, M. G. and Miller, J. P. (1964) Fluvial Processes in Geomorphology. W. H. Freeman, San Francisco.Google Scholar
Mandelbrot, B. B. and Taqqu, M. (1979) Robust R/S analysis of long run serial correlation. 42nd International Statist. Inst., Manila, 1–38.Google Scholar
Mandelbrot, B. B. and Wallis, J. (1968) Noah, Joseph, and operational hydrology. Water Resources Res. 4, 909–918.CrossRefGoogle Scholar
Moran, P. A. P. (1964) On the range of cumulative sums. Ann. Inst. Statist. Math. 16, 109–112.Google Scholar
Newman, C. (1980) Self-similar random fields in mathematical physics. Proc. Measure Theory Conference, Dekalb, Il. Google Scholar
Newman, C. and Wright, L. (1981) An invariance principle for certain dependent sequences. Ann. Prob. 9, 671–675.Google Scholar
Taqqu, M. (1979) Self-similar processes and related ultraviolet and infrared catastrophes. Technical Report No. 423, Cornell University.Google Scholar