Questions about autocorrelation, correlation, and randomness I'm learning about autocorrelation functions and trying to calculate some from a time series. Conceptually, what is the difference between the following two autocorrelation functions?

$$g_{1}(\tau)=\frac{\big\langle (A(t)-\left \langle A \right\rangle)(A(t+\tau)-\left \langle A \right\rangle ) \big\rangle}{\big\langle (A(t) - \left \langle A \right \rangle )^{2} \big \rangle}$$

$$g_{2}(\tau)=\frac{\big\langle A(t)A(t+\tau)\big\rangle}{\big\langle A(t)^{2} \big\rangle}$$

My understanding is that the autocorrelation for the second equation will asymptotically go to zero if there is no correlation between the events in the time series. Is that the case? What about for the first equation? And why?

For the first equation, is the average value $\left \langle A \right \rangle$ the average over all of $\tau$, or do you need to take a running average and calculate it for each time $t$ when doing the calculation?

What happens if the time series is correlated? Will it never approach zero (in the case of the second equation)?

Is there any a priori way of determining how far to take the lag time $\tau$?

Finally, how does all of this relate to randomness?
1) g1 assumes that there is a non-zero mean and that always needs to be subtracted off for the correlation calc to make sense.  g2 assumes that the mean is zero. g2 is incorrect if the mean is not zero. 

2) the mean is taken over the whole sample because the autocorrelation calc has the underlying assumption that the process is stationary so the mean doesn't change during the process.

3) As the lag gets larger, then, if the process is stationary, then the autocorrelations should doe down approach zero. Note that there are "long memory" processes that don't have this characteristic but I don't think that they are
considered stationary processes. Someone feel free to correct me if I'm wrong about long memory processes not being considered stationary.