An observable measured along a Markov chain run has lag-one autocorrelation

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Error, Chance and the Machine, Ch Random walks and Markov chains

Problem

An observable AA measured along a Markov chain run has lag-one autocorrelation ϕ=0.7\phi = 0.7, and the series is well described by an AR(1) process. ((a)) Compute the integrated autocorrelation time τint=(1+ϕ)/[2(1−ϕ)]\tau_{\rm int} = (1+\phi)/[2(1-\phi)]. ((b)) State the factor by which the true error bar of the sample mean exceeds the naive error bar var(A)/n\sqrt{\mathrm{var}(A)/n} that ignores the correlation.

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