I have $n$ successive observation $A_\mu $ of a quantity $A$ and I need to understand how the expectation values of the square of the statistical error depends from the autocorrelation time but a single step in the demonstration is giving me trouble. the author of the book (Monte Carlo simulations in statistical physics -Binder) wrote :
\begin{align*} \langle {(\delta A)^2} \rangle &= \bigg\langle \bigg[\frac{1}{n} \sum_{\mu =1}^n (A_\mu - \langle A \rangle) \bigg]^2 \bigg\rangle \\ &= \frac{1}{n^2} \sum_{\mu=1}^n \langle(A_\mu -\langle A \rangle)^2 \rangle + \frac{2}{n^2}\sum_{ \mu_1= 1}^n \sum_{\mu_2 = \mu_1 +1}^n \bigg(\langle A_{\mu_1} A_{\mu_2} \rangle- \langle A\rangle \bigg) \end{align*}
changing the summation index $\mu_2 $ to $\mu_1 +\mu $ with $ \mu = \mu_2 - \mu_1$this equation can be rewritten as :
$$ \langle {(\delta A)^2} \rangle = \frac{1}{n} \bigg[\langle A^2\rangle - \langle A \rangle ^2 + 2 \sum_{\mu =1}^n \bigg( 1-\frac{\mu}{n} \bigg) \bigg(\langle A_0 A_\mu \rangle - \langle A \rangle \bigg) \bigg] $$
how does this change in the summation index works?
please help me understand this passage