deletions | additions       diff --git a/time_series_convergence.tex b/time_series_convergence.tex  index 7b7d9a0..20f0927 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{Ambegaokar_2010,Frenkel_2002} \cite{Ambegaokar_2010, Frenkel_2002}  it can be shown that the second term on the RHS of 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, 
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\label{eq:delta3}  \Delta_A^2 = \frac{\text{Var}A}{M} (1 + 2 \tau_A)   \end{equation}  where $\xi_A = 1 + 2 \tau_A$ is the \emph{statistical inefficiency} and $M/\xi_A < M$ is the effective number of uncorrelated samples. We can compute the integrated autocorrelation time and hence the statistical inefficiency using the \texttt{pymbar} library. library \cite{Chodera_2007, Shirts_2008}.  We can rearrange Eq.~(\ref{eq:delta3}) to find  \begin{equation}