Variance of partial sums of stationary sequences

Journal article

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 ($0<\gamma<2$).

Authors: Deligiannidis, G; Utev, S

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Institute of Mathematical Statistics
• Journal: Annals of Probability
• Volume: 41
• Issue: 5
• Pages: 3606-3616
• Publication date: 2012-05-18
• DOI: 10.1214/12-AOP772
• ISSN: 0091-1798
• Keywords: long-range dependence; stationary sequences; Fourier analysis; tempered distributions