On the Long-Range Dependence of Fractional Brownian Motion

Abstract

This paper clarifies that the fractional Brownian motion,BH(t), is of long-range dependence (LRD) for the Hurst parameter0<H<1exceptH=1/2. In addition, we note that the fractional Brownian motion is positively correlated for0<H<1exceptH=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 ofX(t). One example is that the differential ofBH(t), in the domain of generalized functions, changes the LRD ofBH(t)to be of short-range dependence (SRD) when0<H<0.5.