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.
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty