Using the definition of the blocking transformation and the distributive property of the covariance, it is clear that since h =|i-j| we can define
\begin{align} \gamma_{k+1}(h) &= cov\left( ({X}_{k+1})_{i}, ({X}_{k+1})_{j} \right) \nonumber \\ &= \frac{1}{4}cov\left( ({X}_{k})_{2i-1} + ({X}_{k})_{2i}, ({X}_{k})_{2j-1} + ({X}_{k})_{2j} \right) \nonumber \\ &= \frac{1}{2}\gamma_{k}(2h) + \frac{1}{2}\gamma_k(2h+1) \hspace{0.1cm} \mathrm{h = 0} \tag{23}\\ &=\frac{1}{4}\gamma_k(2h-1) + \frac{1}{2}\gamma_k(2h) + \frac{1}{4}\gamma_k(2h+1) \quad \mathrm{else} \tag{24} \end{align}The quantity \hat{X} is asymptotic uncorrelated by assumption, \hat{X}_k is also asymptotic uncorrelated. Let's turn our attention to the variance of the sample mean V(\overline{X}) .