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Julian Schrenk added missing citations
over 9 years ago
Commit id: bfd7a95b4c43a2d739f9e20e202951e5e183ad5e
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diff --git a/time_series_convergence.tex b/time_series_convergence.tex
index 56e86fd..5157f0c 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,