Our estimate of \( \mu_{X_i}^{\phantom X} \) is then the sample mean \( \bar x \) itself, in accordance with the the central limit theorem:

$$ \mu_{X_i}^{\phantom X} = \langle x_i\rangle \approx \frac{1}{n}\sum_{k=1}^n x_k = \bar x $$

Using \( \bar x \) in place of \( \mu_{X_i}^{\phantom X} \) we can give an estimate of the covariance in Eq. (13)

$$ \mathrm{cov}(X_i, X_j) = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle \approx\langle (x_i - \bar x)(x_j - \bar{x})\rangle, $$

resulting in

$$ \frac{1}{n} \sum_{l}^n \left(\frac{1}{n}\sum_{k}^n (x_k -\bar x_n)(x_l - \bar x_n)\right)=\frac{1}{n}\frac{1}{n} \sum_{kl} (x_k -\bar x_n)(x_l - \bar x_n)=\frac{1}{n}\mathrm{cov}(x) $$