  The long range dependence of the fractional Brownian motion (fBm), fractional
Gaussian noise (fGn), and differentiated fGn (DfGn) is described by the Hurst
exponent $H$. Considering the realisations of these three processes as time
series, they might be described by their statistical features, such as half of
the ratio of the mean square successive difference to the variance,
$\mathcal{A}$, and the number of turning points, $T$. This paper investigates
the relationships between $\mathcal{A}$ and $H$, and between $T$ and $H$. It is
found numerically that the formulae $\mathcal{A}(H)=a{\rm e}^{bH}$ in case of
fBm, and $\mathcal{A}(H)=a+bH^c$ for fGn and DfGn, describe well the
$\mathcal{A}(H)$ relationship. When $T(H)$ is considered, no simple formula is
found, and it is empirically found that among polynomials, the fourth and
second order description applies best. The most relevant finding is that when
plotted in the space of $(\mathcal{A},T)$, the three process types form
separate branches. Hence, it is examined whether $\mathcal{A}$ and $T$ may
serve as Hurst exponent indicators. Some real world data (stock market indices,
sunspot numbers, chaotic time series) are analyzed for this purpose, and it is
found that the $H$'s estimated using the $H(\mathcal{A})$ relations (expressed
as inverted $\mathcal{A}(H)$ functions) are consistent with the $H$'s extracted
with the well known wavelet approach. This allows to efficiently estimate the
Hurst exponent based on fast and easy to compute $\mathcal{A}$ and $T$, given
that the process type: fBm, fGn or DfGn, is correctly classified beforehand.
Finally, it is suggested that the $\mathcal{A}(H)$ relation for fGn and DfGn
might be an exact (shifted) $3/2$ power-law.
