Variance of partial sums of stationary sequences

Let $X_1,X_2,\dots$ be a centred sequence of weakly stationary random variables with spectral measure $F$ and partial sums $S_n=X_1+\cdots +X_n$. We show that $\mathrm{var}(S_n)$ is regularly varying of index $\gamma$ at infinity, if and only if $G(x):={\int }_{-x}^{x}F(\mathrm{d}x)$ is regularly varying of index $2-\gamma$ at the origin ($0<\gamma <2$).