Error in histogram measurements

I ran into the following statement here and here but I believe it's more general.

Let's suppose we're running a simulation of a system and we are interested in the distribution of a quantity (say $$M$$). One would then run the simulation for a long time and construct a histogram of $$M$$. The counts in the histogram corresponding to a value of $$M$$ are $$N(M)$$. One can denote by $$\langle N(M) \rangle$$ the ensemble average of $$N(M)$$, over all simulations of a given length. The statement is that the error in the histogram goes like:

$$\delta^2N(M) = (1+2 \tau) \langle N(M) \rangle$$

Here $$\tau$$ is the correlation time of the system, or, of the quantity of interest. My first question is, is $$\delta^2N(M)$$ the variance or the standard deviation from the ensemble average? Second, how would you prove the above relation?

As an example, one could think of a Monte Carlo simulation of an Ising system, trying to obtain a distribution of the magnetization.

• Take a look at Appendix B of this paper. – user8153 Apr 2 at 0:05
• @user8153, almost there. The precise step I need is cited from a paper I have no access to, but it's a good starting point. Thank you. – Botond Apr 19 at 17:58