Is there any result like the [Berry-Esseen theorem](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem) - which says that the sample mean of a random sample converges to the true mean at a rate of $\frac{1}{\sqrt{n}}$ - that would apply to the observed autocorrelation? I.e. how fast would the observed autocorrelation in a sample converge to the true autocorrelation (say, an AR(x) process)?

_This question sprung from a minor point around [this answer](https://stats.stackexchange.com/a/314014/182174) (rate of convergence of the observed autocorrelation vs rate of convergence of the confidence band in an ACF - we do know the latter) but I thought it was sufficiently general to merit its own question._