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}