We now define blocking transformations. The idea is to take the mean of subsequent pair of elements from \vec{X} and form a new vector \vec{X}_1 . Continuing in the same way by taking the mean of subsequent pairs of elements of \vec{X}_1 we obtain \vec{X}_2 , and so on. Define \vec{X}_i recursively by:
\begin{align} (\vec{X}_0)_k &\equiv (\vec{X})_k \nonumber \\ (\vec{X}_{i+1})_k &\equiv \frac{1}{2}\Big( (\vec{X}_i)_{2k-1} + (\vec{X}_i)_{2k} \Big) \qquad \text{for all} \qquad 1 \leq i \leq d-1 \tag{22} \end{align}The quantity \vec{X}_k is subject to k blocking transformations. We now have d vectors \vec{X}_0, \vec{X}_1,\cdots,\vec X_{d-1} containing the subsequent averages of observations. It turns out that if the components of \vec{X} is a stationary time series, then the components of \vec{X}_i is a stationary time series for all 0 \leq i \leq d-1
We can then compute the autocovariance, the variance, sample mean, and number of observations for each i . Let \gamma_i, \sigma_i^2, \overline{X}_i denote the autocovariance, variance and average of the elements of \vec{X}_i and let n_i be the number of elements of \vec{X}_i . It follows by induction that n_i = n/2^i .