Variance of partial sums of stationary sequences

Abstract

Let $X_{1},X_{2},\ldots$ 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 $\operatorname{var} (S_{n})$ 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 ($0<\gamma<2$).