On the long-range dependence of fractional Brownian motion

Abstract

This paper clarifies that the fractional Brownian motion, B H (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 B H (t), in the domain of generalized functions, changes the LRD of B H (t) to be of short-range dependence (SRD) when 0 < H < 0.5. © 2013 Ming Li.