Flyvbjerg and Petersen demonstrated that the sequence \( \{e_k\}_{k=0}^{d-1} \) is decreasing, and conjecture that the term \( e_k \) can be made as small as we would like by making \( k \) (and hence \( d \)) sufficiently large. The sequence is decreasing. It means we can apply blocking transformations until \( e_k \) is sufficiently small, and then estimate \( \mathrm{var}(\overline{X}) \) by \( \widehat{\sigma}^2_k/n_k \).
For an elegant solution and proof of the blocking method, see the recent article of Marius Jonsson (former MSc student of the Computational Physics group).