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I am trying to calculate the variance: $$ \langle(\bar{O}-<O>)^2\rangle $$ of the Monte-Carlo estimator $$ \bar{O}=\frac{1}{M}\sum_{m=1}^M{O_m} $$ For uncorrelated samples.

In order to do so, I need to know the expression for the covariance between two samples (so I can use: $Cov(O_m,O_n)=0$) but looking at definitions of sample covariance online only confused me, since it all seems to assume two different pools of samples, and gives the "total" covariance between these two pools rather than the expression for covariances between two individual samples.

Roger V.
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Nitzan R
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    This is a strange-sounding question because it seems to confuse a formula for the variance of data with formulas for computing variances of linear combinations of random variables. Maybe we can clear it up if you would tell us what a "sample" is. – whuber Dec 02 '22 at 14:19
  • What I am looking for, in essence, is how the fact that the samples (which I think should mean data points) are uncorrelated comes into play when attempting to calculate the variance of the monte-carlo estimator. – Nitzan R Dec 02 '22 at 14:51
  • I see: you are looking for the standard error of this mean. This thread might help: https://stats.stackexchange.com/questions/495792. – whuber Dec 02 '22 at 15:15

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