ALF-QMC · ALF-Import-Bot · Jul 25, 2025 · Mar 29, 2026 · Copilot · Mar 29, 2026
diff --git a/Documentation/sampling.tex b/Documentation/sampling.tex
@@ -74,7 +74,7 @@ \subsection{An explicit example of error estimation}\label{sec:autocorr}
 In order to determine the autocorrelation time, we calculate the correlation function
 \begin{equation}
 \label{eqn:autocorrel}
-	S_{\hat{O}}(t_{\textrm{Auto}})=\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\frac{\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle \right)}{\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i}-\left\langle \hat{O}\right\rangle \right)}\, ,
+	S_{\hat{O}}(t_{\textrm{Auto}})=\frac{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle \right)}{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i}-\left\langle \hat{O}\right\rangle \right)}\, ,
 \end{equation}
 where $O_i$ refers to the Monte Carlo estimate of the observable $\hat{O}$ in the $i^{\text{th}}$ bin. This function typically shows an exponential decay and the decay rate defines the autocorrelation time.
-	S_{\hat{O}}(t_{\textrm{Auto}})=\frac{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle \right)}{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i}-\left\langle \hat{O}\right\rangle \right)}\, ,
-\end{equation}
-where $O_i$ refers to the Monte Carlo estimate of the observable $\hat{O}$ in the $i^{\text{th}}$ bin. This function typically shows an exponential decay and the decay rate defines the autocorrelation time.
+	S_{\hat{O}}(t_{\textrm{Auto}})=\frac{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} \right)}{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} \right)\left(O_{i}-\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} \right)}\, ,
+\end{equation}
+where $O_i$ refers to the Monte Carlo estimate of the observable $\hat{O}$ in the $i^{\text{th}}$ bin, and the lag-dependent mean is defined as
+\begin{equation}
+	\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} = \frac{1}{N_{\textrm{Bin}}-t_{\textrm{Auto}}} \sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}} O_i \, .
+\end{equation}
+This function typically shows an exponential decay and the decay rate defines the autocorrelation time.
-	S_{\hat{O}}(t_{\textrm{Auto}})=\frac{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle \right)}{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i}-\left\langle \hat{O}\right\rangle \right)}\, ,
-\end{equation}
-where $O_i$ refers to the Monte Carlo estimate of the observable $\hat{O}$ in the $i^{\text{th}}$ bin. This function typically shows an exponential decay and the decay rate defines the autocorrelation time.
+	S_{\hat{O}}(t_{\textrm{Auto}})=\frac{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} \right)}{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} \right)\left(O_{i}-\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} \right)}\, ,
+\end{equation}
+where $O_i$ refers to the Monte Carlo estimate of the observable $\hat{O}$ in the $i^{\text{th}}$ bin, and the lag-dependent mean is defined as
+\begin{equation}
+	\left\langle \hat{O}\right\rangle_{t_{\textrm{Auto}}} = \frac{1}{N_{\textrm{Bin}}-t_{\textrm{Auto}}} \sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}} O_i \, .
+\end{equation}
+This function typically shows an exponential decay and the decay rate defines the autocorrelation time.
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