Abstract

In 1950 H. E. Hurst published the results of his investigations of water out ow from the great lakes of the Nile basin. Hurst wanted to determine the reservoir capacity that would be needed to develop the irrigation along the Nile to its fullest extent. His work motivated the notion of long range dependence through the application of a statistic that he developed for his study. This is the rescaled range statistic-the R=S statistic. Given data Xi, i = 1; : : : ; n, set n = 1 n Pn i=1 Xi; and let S i = Pi j=1 (Xj 􀀀 n), for i = 1; : : : ; j; M n = max (0; S 1 ; : : : ; Sn ) and m n = min (0; S 1 ; : : : ; Sn ) : Dene the adjusted range Rn = M n 􀀀m n: The rescaled range statistic (the R=S statistic) is Rn = M n 􀀀m n: where Dn is the sample standard deviation Dn = qPn i=1 (Xi 􀀀 n)2 =n, for n  1: Hurst argued via a small simulation study that if Xi, i = 1; : : : ; n, are i.i.d. normal then (R=S)n should grow in the order of p n. (Hurst was later proved correct by Feller.) However, Hurst found that for the Nile River data, (R=S)n increased not in the order of p n; but in the order nH, where H ranged between :68 and :80 with a mean of :75. For annual tree ring data H ranged between :79 and :86 with a mean of :80, for sunspots and wheat prices an average H of :69 was obtained, and data on the thickness of annual layers of lake mud deposits gave an average of H = :69. All of the above data had normal-like histograms, yet all gave estimates of H consistently greater than 1=2, which an i.i.d. normal model would give. This is now called the Hurst phenomenon. We shall discuss 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.