We now define the blocking transformations. The idea is to take the mean of subsequent pair of elements from \( \boldsymbol{X} \) and form a new vector \( \boldsymbol{X}_1 \). Continuing in the same way by taking the mean of subsequent pairs of elements of \( \boldsymbol{X}_1 \) we obtain \( \boldsymbol{X}_2 \), and so on. Define \( \boldsymbol{X}_i \) recursively by:
$$ \begin{align} (\boldsymbol{X}_0)_k &\equiv (\boldsymbol{X})_k \nonumber \\ (\boldsymbol{X}_{i+1})_k &\equiv \frac{1}{2}\Big( (\boldsymbol{X}_i)_{2k-1} + (\boldsymbol{X}_i)_{2k} \Big) \qquad \text{for all} \qquad 1 \leq i \leq d-1 \tag{1} \end{align} $$