A central limit theorem is proved for the sample serial covariances of an ergodic, stationary, purely nondeterministic process whose linear innovations have their first four moments as for a sequence of independent random variables. The necessary and sufficient condition for the theorem is then that the spectra be square integrable.
"The Asymptotic Distribution of Serial Covariances." Ann. Statist. 4 (2) 396 - 399, March, 1976. https://doi.org/10.1214/aos/1176343415