The quantity \( \boldsymbol{X}_k \) is subject to \( k \) blocking transformations. We now have \( d \) vectors \( \boldsymbol{X}_0, \boldsymbol{X}_1,\cdots,\boldsymbol{X}_{d-1} \) containing the subsequent averages of observations. It turns out that if the components of \( \boldsymbol{X} \) is a stationary time series, then the components of \( \boldsymbol{X}_i \) is a stationary time series for all \( 0 \leq i \leq d-1 \)
We can then compute the autocovariance (or just covariance), the variance, sample mean, and number of observations for each \( i \). Let \( \gamma_i, \sigma_i^2, \overline{X}_i \) denote the covariance, variance and average of the elements of \( \boldsymbol{X}_i \) and let \( n_i \) be the number of elements of \( \boldsymbol{X}_i \). It follows by induction that \( n_i = n/2^i \).