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The long range dependence of the fractional Brownian motion (fBm), fractionalGaussian noise (fGn), and differentiated fGn (DfGn) is described by the Hurstexponent $H$. Considering the realisations of these three processes as timeseries, they might be described by their statistical features, such as half ofthe ratio of the mean square successive difference to the variance,$\mathcal{A}$, and the number of turning points, $T$. This paper investigatesthe relationships between $\mathcal{A}$ and $H$, and between $T$ and $H$. It isfound numerically that the formulae $\mathcal{A}(H)=a{\rm e}^{bH}$ in case offBm, 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 isfound, and it is empirically found that among polynomials, the fourth andsecond order description applies best. The most relevant finding is that whenplotted in the space of $(\mathcal{A},T)$, the three process types formseparate branches. Hence, it is examined whether $\mathcal{A}$ and $T$ mayserve as Hurst exponent indicators. Some real world data (stock market indices,sunspot numbers, chaotic time series) are analyzed for this purpose, and it isfound that the $H$'s estimated using the $H(\mathcal{A})$ relations (expressedas inverted $\mathcal{A}(H)$ functions) are consistent with the $H$'s extractedwith the well known wavelet approach. This allows to efficiently estimate theHurst exponent based on fast and easy to compute $\mathcal{A}$ and $T$, giventhat the process type: fBm, fGn or DfGn, is correctly classified beforehand.Finally, it is suggested that the $\mathcal{A}(H)$ relation for fGn and DfGnmight be an exact (shifted) $3/2$ power-law.

On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points