The Hurst phenomenon and the rescaled range statistic

In his 1951 study of Nile River data, H.E. Hurst introduced the rescaled range statistic-the R/S statistic. He argued via a small simulation study that if Xi, i=1,…,n, are i.i.d. normal then the R/S statistic should grow in the order of n. However, Hurst found that for the Nile River data, the R/S statistic increased not in the order of n, but in the order nH, where H ranged between 0.75 and 0.80. He discovered that the effect also appeared in other sets of data. This is now called the Hurst phenomenon. We shall establish some unexpected universal asymptotic properties of the R/S statistic, which show conclusively that the Hurst phenomenon can never appear for i.i.d. data.