If we generate white noise and do the FFT of it, we get the same amplitude for each of the frequencies. Therefore, the output of the FFT of the noise follows approximately the power law $FFT(f)=m\frac{1}{f^\alpha}$ with $\alpha=0$ and $m$ a constant.
On the other hand if we generate what is called as pink noise then $\alpha=1$. There is also the case where we generate brown noise, that is noise generated through Brownian motion. As Brownian motion remembers its previous state, we can say that it has 'memory'. In the case of Brown noise it can be checked that $\alpha\approx2$.
My question is: If we analyse some noise and get that the power spectrum (FFT) follows approximately a power law with, lets say, $\alpha=3$, can we say anything about how much memory the noise contains?