Correlation Time

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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.