formula

${\rm V}[X_N^{(2)}] &=& \frac{X_0^2}{N^2} \left\{ 2\sum_{j>i} \mu^{N - j + 1}\mu^{N - i + 1} + \sum_{i=1}^N \mu^i -\left( \sum_{i=1}^N \mu^i \right)^2 \right\} \\ &=& \frac{X_0^2}{N^2} \left\{ 2\sum_{j<i} \mu^{i}\mu^{j} + \sum_{i=1}^N \mu^i -\left( \sum_{i=1}^N \mu^i \right)^2 \right\} \\ &=& 0$