I know that for standard Brownian motion, the total variation sampled at intervals of length $\Delta$ converges to $V(\Delta) = C \Delta^{-1/2}$ for some constant $C$. I wish to use this fact to study whether or not my data behaves as Brownian motion by calculating $V(\Delta)$ for many $\Delta$ in order to estimate $\beta$ in $V(\Delta) = C \Delta ^{-\beta}$, and verify that it is close to $\frac{1}{2}$.


One thing that I'm unsure about however, is how this total variation would behave if it is not standard Brownian motion. For example, if it is fractal Brownian motion with exponent $\alpha$, would I expect that my estimated $\hat{\beta}$ be close to $\alpha$ (or $1- \alpha$)? Or does this not apply to fractal brownian motion in general?


What is some good literature that discusses the behavior total variation of fractals (and specifically fractal Brownian motion)?


I have tried googling extensively but it always just leads me to normal Brownian motion.



Literature on the total variation of fractal graphs/fractal Brownian motion?
I presume that you are considering sample paths on the time interval $[0,1]$, so that the number of samples is $n=1/\Delta$. The fractional Brownian motion has stationary increments, whence
$$
\mathbf E V(\Delta) = n \mathbf E |Z(\Delta)| ;. 
$$
Since $Z(\Delta)$ is Gaussian with variance $\Delta^{2\alpha}$,
$$
\mathbf E |Z(\Delta)| = C \Delta^\alpha
$$
for a constant $C$, and therefore
$$
\mathbf E V(\Delta) = C \Delta^{\alpha-1} \;.
$$


