Is there any result like the Berry-Esseen 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 (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.

How fast would the observed autocorrelation in a sample converge to the true autocorrelation (say, an AR(x) process)? Are there any results like the Central Limit Theorem - which says that the sample mean of a random sample converges to the true mean at a rate of $\frac{1}{\sqrt{n}}$ - or the Berry-Esseen theorem that would apply to the observed autocorrelation?

This question sprung from a minor point around this answer (rate of convergence of the observed autocorrelation vs rate of convergence of the confidence band in an ACF - we do know the latter) and I thought it was sufficiently general to merit its own question.