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 $G(x):=\int_{-x}^xF(\mathrm {d}x)$ is regularly varying of index $2-\gamma$ at the origin ($0<\gamma<2$).