Cross-posted on Stats.SE and on MMSE.

In the case of Monte Carlo simulations:

  • Autocorrelation Time ($\tau_{\text{int}}$): A measure of how many steps are needed for the correlations in the chain to become small; it affects the variance of estimators based on the Markov chain.

  • Relaxation Time ($\tau_{\text{relax}}$): The time it takes for the Markov chain to approach its equilibrium distribution from an arbitrary starting point.

  • Longest Relaxation Time ($\tau_1$): The slowest timescale on which the system relaxes to equilibrium, which is critical for understanding the overall convergence rate of the Markov chain.

I realize that autocorrelation time is the sum of autocorrelation values over a given maximum lag time. So, the formula becomes:

$$ \tau_{\text{int}} = \frac{1}{2} + \sum_{t=1}^{\text{max-lag}} ACF(t) $$

where $ACF()$ is the autocorrelation function:

$$ ACF(k) = \frac{1}{(n-k)\sigma^2} \sum_{i=1}^{n-k} (x_i - \mu)(x_{i+k} - \mu) $$

I also realize that, relaxation time can be found by fitting an autocorrelation value to the exponentia decay function ($A_0 \cdot e^{\frac{-t}{\tau_\text{relax}}}$).

How can I compute the longest relaxation time?

  • $\begingroup$ Please put a banner at the top of your question that has a link to all cross-postings, not just the MMSE one but also the Stats one. $\endgroup$ Mar 2 at 12:17
  • $\begingroup$ Is the longest relaxation time actually well defined? It was my understanding that Monte Carlo Methods will never converge to a single defined value due to the randomness of the samples. Isn't the longest relaxation time simply the maximum time (or samples) which were considered in the Monte Carlo Simulation? $\endgroup$
    – MPIchael
    Mar 4 at 14:50
  • 1
    $\begingroup$ @MPIchael, you tell me. $\endgroup$
    – user366312
    Mar 4 at 16:56


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