I am reading Ernie Chan's blog post "Mean reversion, momentum, and volatility term structure". It says that
To be precise, if $z$ is the log price, then volatility, sampled at intervals of $\tau$, is $$\text{Volatility}(\tau)=\sqrt{\text{Var}(z_t-z_{t-\tau})}$$ where $\text{Var}$ means taking the variance over many sample times.
In general, we can write $$\text{Var}(z_t-z_{t-\tau})\propto \tau^{2H}$$ where H is called the "Hurst exponent", and it is equal to 0.5 for a true geometric random walk, but will be less than 0.5 for mean reverting prices, and greater than 0.5 for trending prices.
It looks correct intuitively, as for a trending price, the variance increases faster as the sample interval gets larger, and for a mean reverting price, the variance doesn't change that much accordingly.
Can anyone give some explanation (or proof) on that (and why it is equal to 0.5 for a true geometric random walk)? Thanks!