Given a fBm $B_H(t)$, we define the fGn $G(j)$ as $$ G(j) = B_H(j+1)-B_H(j) \quad j\in\mathbb{N} \ . $$ If $H=\frac12$, then $B_H(t)$ is the usual brownian motion, and $G(j)$ is a (discrete) white noise. This noise is identically distributed for all $j$, so it would have a Hurst exponent equal to $0$, but I've seen some people say it is $-1/2$. Why is that?
And what about the Hurst exponent of the fGn corresponding to some $B_H(t)$ with $H\in(0,1)$? What would its value be?