In Markov Chain Monte Carlo (MCMC) simulations, the thermal equilibriaquantities are estimated by ensemble average over a sample set containing alarge number of correlated samples. These samples are selected in accordancewith the probability distribution function, known from the partition functionof equilibrium state. As the stochastic error of the simulation results issignificant, it is desirable to understand the variance of the estimation byensemble average, which depends on the sample size (i.e., the total number ofsamples in the set) and the sampling interval (i.e., cycle number between twoconsecutive samples). Although large sample sizes reduce the variance, theyincrease the computational cost of the simulation. For a given CPU time, thesample size can be reduced greatly by increasing the sampling interval, whilehaving the corresponding increase in variance be negligible if the originalsampling interval is very small. In this work, we report a few general rulesthat relate the variance with the sample size and the sampling interval. Theseresults are observed and confirmed numerically. These variance rules arederived for the MCMC method but are also valid for the correlated samplesobtained using other Monte Carlo methods. The main contribution of this workincludes the theoretical proof of these numerical observations and the set ofassumptions that lead to them.
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