I am currently reading the paper "Fractional Brownian Motions, Fractional Noises and Applications" from Mandelbrot and Van Ness (https://users.math.yale.edu/mandelbrot/web_pdfs/052fractionalBrownianMotions.pdf) and I am struggling with some of the things pointed out with regard to the empirical foundation for the fractional brownian motion. In particular on page 258 he defines the sequential range to be $$M(t,T, \omega) = \sup\limits_{t\leq s\leq t+T} (X(s,\omega)-X(t,\omega) - \inf\limits_{t\leq s\leq t+T} (X(s,\omega)-X(t,\omega).$$ On page 259 et seq. he then continues with the empirical result of Hurst that this range of certain water flows in rivers is rather proportional to $T^H$ for some $1/2 < H < 1$. Mandelbrot/Van Ness then say that this is a surprise as in typical models of the form $$X(t, \omega) = \int_0^t Y(s,\omega) \mathrm{d}s$$ where $Y$ is an integrable covariance function, you would expect the sequential range to be proportional to $\sqrt{T}$. However, unfortunately I cannot follow this point and lack the intuition both for the model and for why this is surprising and important.
I am very new to to this field and would appreciate any tip on how to see this and especially some intuition behind it!
Best regards and many thanks in advance!