THE HURST EFFECT - THE SCALE OF FLUCTUATION APPROACH

After more than 40 years the so-called Hurst effect remains an open pr oblem in stochastic hydrology. Historically, its existence has been ex plained either by preasymptotic behavior of the rescaled adjusted rang e R(n), certain classes of nonstationarity in time series, infinite m emory, or erroneous estimation of the Hurst exponent. Various statisti cal tests to determine whether an observed time series exhibits the Hu rst effect are presented. The tests are based on the fact that for the family of processes in the Brownian domain of attraction, R(n)/((the tan))1/2 converges in distribution to a nondegenerate random variable with known distribution (functional central limit theorem). The scale of fluctuation theta, defined as the sum of the correlation function, plays a key role. Application of the tests to several geophysical time series seems to indicate that they do not exhibit the Hurst effect, a lthough those series have been used as examples of its existence, and furthermore the traditional pox diagram method to estimate the Hurst e xponent gives values larger than 0.5. It turned out that the coefficie nt in the relation of R(n) versus n, which is directly proportional t o the scale of fluctuation, was more important than the exponent. The Hurst effect motivated the popularization of 1/f noises and related id eas of fractals and scaling. This work illustrates how delicate the pr ocedures to deal with infinity must be.