Open Access
September 2013 Variance of partial sums of stationary sequences
George Deligiannidis, Sergey Utev
Ann. Probab. 41(5): 3606-3616 (September 2013). DOI: 10.1214/12-AOP772


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 ($0<\gamma<2$).


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George Deligiannidis. Sergey Utev. "Variance of partial sums of stationary sequences." Ann. Probab. 41 (5) 3606 - 3616, September 2013.


Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1291.60068
MathSciNet: MR3127893
Digital Object Identifier: 10.1214/12-AOP772

Primary: 60G10
Secondary: 42A24

Keywords: Fourier analysis , long-range dependence , Stationary sequences , tempered distributions

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 5 • September 2013
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