Autocorrelation time

An autocorrelation time measures how long an observable retains dependence on its past. It depends on both the process and the chosen observable. For a stationary sequence with finite positive variance and autocorrelation function $\rho (k)$, the integrated autocorrelation time is ${\tau }_{\mathrm{int}}=1+2{\sum }_{k\ge 1}\rho (k)$ when the series is absolutely summable. It controls the leading variance of a long sample mean.

An exponential autocorrelation time instead describes tail decay, for example $\rho (k)\approx C{e}^{-k/{\tau }_{\exp }}$. These are different summaries. A stationary autoregressive process of order one with coefficient $0<a<1$ has $\rho (k)=a^k$, so ${\tau }_{\mathrm{int}}=(1+a)/(1-a)$ and ${\tau }_{\exp }=-1/\log a$. For a Markov chain with several decay modes, different observables can emphasize different timescales.