Prove Eq. (1.16) and discuss the increase in the variance of sample mean with respect to the independent case when the series has a non-zero first-order autocorrelation coefficient and zero autocorrelations for all higher lags. That is, the series follows an MA(1) model.

\(\boldsymbol{\Gamma}(0)=\left[\begin{array}{cccc}
\gamma_{11}(0) & \gamma_{12}(0) & \cdots & \gamma_{1 k}(0) \\
\gamma_{21}(0) & \gamma_{22}(0) & \cdots & \gamma_{2 k}(0) \\
\vdots & \vdots & \ddots & \vdots \\
\gamma_{k 1}(0) & \gamma_{k 2}(0) & \cdots & \gamma_{k k}(0)
\end{array}\right] \tag{1.6}\)