= 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 $\tau_{\mathrm{int}}=1+2\sum_{k\geq1}\rho(k)$ 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 $\rho(k)\approx Ce^{-k/\tau_{\mathrm{exp}}}$. 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 $\tau_{\mathrm{exp}}=-1/\log a$. For a Markov chain with several decay modes, different observables can emphasize different timescales.
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