Correlation Time

In the case of Gaussian white noise, since the correlation function is a delta function, the correlation time is zero. In the case of red noise the autocorrelation function is a negative exponential $\langle X(t)X(t+\tau )\rangle =\exp({\frac {-\tau }{t_{c}}})$ , so that $\tau =t_{c}$ is the correlation time, because that’s how long it takes to drop by an order of magnitude. In some sense a longer correlation time corresponds to a longer “memory” of the previous path.