The Hurst effect under trends

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 formf(n) =c(m+n)^ß, where m is an arbitrary non-negative parameter andcis not 0. For – ½ <ß< 0 the Hurst exponent is shown to be precisely given by 1 +ß.Forß≦ – ½ and forß= 0 the Hurst exponent is 0.5, while forß > 0it 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.