Let $r(j)$ denote the $j$th autocorrelation based on a sample of $N$ consecutive observations on a stationary linear stochastic process. Under mild regularity conditions on the process, an iterated logarithm result is given for the convergence of $r(j)$ as $N \rightarrow \infty$ to the corresponding process autocorrelation $\rho (j)$.
"An Iterated Logarithm Result for Autocorrelations of a Stationary Linear Process." Ann. Probab. 2 (2) 328 - 332, April, 1974. https://doi.org/10.1214/aop/1176996714