On the Long-Range Dependence of Fractional Brownian Motion

This paper clarifies that the fractional Brownian motion, BH(t), is of long-range dependence (LRD) for the Hurst parameter 0<H<1 except H=1/2. In addition, we note that the fractional Brownian motion is positively correlated for 0<H<1 except H=1/2. Moreover, we present a theorem to state that the differential or integral of a random function, X(t), may substantially change the statistical dependence of X(t). One example is that the differential of BH(t), in the domain of generalized functions, changes the LRD of BH(t) to be of short-range dependence (SRD) when 0<H<0.5