deletions | additions       diff --git a/time_series_convergence.tex b/time_series_convergence.tex  index bd7d072..42883e4 100644  --- a/time_series_convergence.tex  +++ b/time_series_convergence.tex 
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\begin{equation}  \langle A_i A_j\rangle = \langle A_i \rangle \langle A_j \rangle = \langle A \rangle^2,   \end{equation}  and the second term in Eq.~\ref{eq:delta1} vanishes. Since we are exploring the basin using a MCMC our samples are far from being independent and a more accurate estimate for the statistical error can be derived under the assumption of fast decay as $|i-j|\rightarrow \infty$. With some algebra \cite{Ambeogacar} it can be shown that the second term on the RHS of Eq.~\ref{eq:delta} Eq.~\ref{eq:delta1}  is equal to \begin{equation}  \label{eq:delta2}  \frac{1}{M^2}\sum_{i \neq j = 1}^M (\langle A_i A_j\rangle = \langle A \rangle^2) \approx \frac{2}{M}\sum_{t = 1}^{\infty}(\langle A_1 A_{1+t}\rangle- \langle A\rangle^2) \equiv \frac{2}{M} \text{Var}A \tau_A,