Darling-Erdős-type theorems for sums of Gaussian variables with long-range dependence (Q1272158)

The paper generalizes the result of \textit{D. A. Darling} and \textit{P. Erdős} [Duke Math. J. 23, 143-155 (1956; Zbl 0070.13806)]. The authors prove a similar result for the Gaussian stationary random sequence with controlled decay of covariances. Provided some restriction on the slowly varying function \(\ell\) they assume either \(\sum _{i,j=1}^n r(i-j)=n^{\alpha}{\ell}(n)\) or \(r(n)=n^{-\alpha}{\ell}(n)\), where \(r\) is the covariance function and \(0<\alpha <2\).