The Hurst Effect: The scale of fluctuation approach

After more than 40 years the so‐called Hurst effect remains an open problem in stochastic hydrology. Historically, its existence has been explained either by preasymptotic behavior of the rescaled adjusted range , certain classes of nonstationarity in time series, infinite memory, or erroneous estimation of the Hurst exponent. Various statistical tests to determine whether an observed time series exhibits the Hurst effect are presented. The tests are based on the fact that for the family of processes in the Brownian domain of attraction, converges in distribution to a nondegenerate random variable with known distribution (functional central limit theorem). The scale of fluctuation θ, 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, although those series have been used as examples of its existence, and furthermore the traditional pox diagram method to estimate the Hurst exponent gives values larger than 0.5. It turned out that the coefficient in the relation of versus n , which is directly proportional to the scale of fluctuation, was more important than the exponent. The Hurst effect motivated the popularization of 1/ f noises and related ideas of fractals and scaling. This work illustrates how delicate the procedures to deal with infinity must be.