In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process BH(t) on [0, T], that starts at zero, has expectation zero for all t in [0, T],
It is not clear what you mean by " What other comparisons can be made ..."
Perhaps the most useful other property of this theoretical process is the spectrum. This shows both:
(a) a high (infinite) peak at zero frequency (which relates to "long-range dependence")
(b) a long tail at high frequencies (which relates to smoothness and differentiability)
It then becomes interesting to examine processes with spectra that have only one of these effects.
A different idea is to look at the related concept of fractional-difference processes for discrete-time time-series, and how these relate to the continuous-time processes.
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