One should be careful with times close to \( t_{\mathrm{max}} \), the upper limit of the sums becomes small and we end up integrating over a rather small time interval. This means that the statistical error in \( \phi(t) \) due to the random nature of the fluctuations in \( \mathbf{M}(t) \) can become large.

One should therefore choose \( t \ll t_{\mathrm{max}} \).

Note that the variable \( \mathbf{M} \) can be any expectation values of interest.

The time-correlation function gives a measure of the correlation between the various values of the variable at a time \( t' \) and a time \( t'+t \). If we multiply the values of \( \mathbf{M} \) at these two different times, we will get a positive contribution if they are fluctuating in the same direction, or a negative value if they fluctuate in the opposite direction. If we then integrate over time, or use the discretized version of, the time correlation function \( \phi(t) \) should take a non-zero value if the fluctuations are correlated, else it should gradually go to zero. For times a long way apart the different values of \( \mathbf{M} \) are most likely uncorrelated and \( \phi(t) \) should be zero.