  To model a given time series $F(t)$ with fractal Brownian motions (fBms), it
is necessary to have appropriate error assessment for related quantities.
Usually the fractal dimension $D$ is derived from the Hurst exponent $H$ via
the relation $D=2-H$, and the Hurst exponent can be evaluated by analyzing the
dependence of the rescaled range $\langle|F(t+\tau)-F(t)|\rangle$ on the time
span $\tau$. For fBms, the error of the rescaled range not only depends on data
sampling but also varies with $H$ due to the presence of long term memory. This
error for a given time series then can not be assessed without knowing the
fractal dimension. We carry out extensive numerical simulations to explore the
error of rescaled range of fBms and find that for $0<H<0.5$, $|F(t+\tau)-F(t)|$
can be treated as independent for time spans without overlap; for $0.5<H<1$,
the long term memory makes $|F(t+\tau)-F(t)|$ correlated and an approximate
method is given to evaluate the error of $\langle|F(t+\tau)-F(t)|\rangle$. The
error and fractal dimension can then be determined self-consistently in the
modeling of a time series with fBms.
