Variance of partial sums of stationary sequences

TL;DR

This paper characterizes the asymptotic behavior of the variance of partial sums of stationary sequences, linking it to the regular variation of the spectral measure, thus providing insights into the dependence structure of such sequences.

Contribution

It establishes a precise equivalence between the regular variation of the variance of partial sums and the spectral measure's behavior near zero.

Findings

Variance of partial sums is regularly varying with index γ

Spectral measure's integral is regularly varying with index 2−γ

Provides a characterization of dependence in 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 ($0<\gamma<2$).