### 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) equals c(m plus n)** beta , where m is an arbitrary non-negative parameter and c is not 0. For minus one-half less than beta less than 0 the Hurst exponent is shown to be precisely given by 1 plus beta . For beta less than equivalent to minus one-half and for beta equals 0 the Hurst exponent is 0. 5, while for beta greater than 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.

Original language | English (US) |
---|---|

Pages (from-to) | 649-662 |

Number of pages | 14 |

Journal | Journal of Applied Probability |

Volume | 20 |

Issue number | 3 |

State | Published - Sep 1983 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Journal of Applied Probability*,

*20*(3), 649-662.

**HURST EFFECT UNDER TRENDS.** / Bhattacharya, Rabindra N; Gupta, Vijay K.; Waymire, Ed.

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 20, no. 3, pp. 649-662.

}

TY - JOUR

T1 - HURST EFFECT UNDER TRENDS.

AU - Bhattacharya, Rabindra N

AU - Gupta, Vijay K.

AU - Waymire, Ed

PY - 1983/9

Y1 - 1983/9

N2 - 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) equals c(m plus n)** beta , where m is an arbitrary non-negative parameter and c is not 0. For minus one-half less than beta less than 0 the Hurst exponent is shown to be precisely given by 1 plus beta . For beta less than equivalent to minus one-half and for beta equals 0 the Hurst exponent is 0. 5, while for beta greater than 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.

AB - 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) equals c(m plus n)** beta , where m is an arbitrary non-negative parameter and c is not 0. For minus one-half less than beta less than 0 the Hurst exponent is shown to be precisely given by 1 plus beta . For beta less than equivalent to minus one-half and for beta equals 0 the Hurst exponent is 0. 5, while for beta greater than 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.

UR - http://www.scopus.com/inward/record.url?scp=0020809572&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0020809572&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0020809572

VL - 20

SP - 649

EP - 662

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 3

ER -