### 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) |
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Pages (from-to) | 649-662 |

Number of pages | 14 |

Journal | Journal of Applied Probability |

Volume | 20 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1983 |

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty

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## Cite this

*Journal of Applied Probability*,

*20*(3), 649-662. https://doi.org/10.1017/S0021900200023895