Thanks for the answer to this question by Demetris Koutsoyiannis
I think that statistical predictions tend always to the mean as time increases. If we use the maximum entropy principle to obtain these predictions, the result depends on the time scale of entropy maximization. For instance, if the entropy maximization is done on the observation time scale, then the prediction may be equivalent to a prediction obtained by a Markov model. However, other settings of entropy maximization (on several time scales) result in long range dependence (as I have demonstrated in my 2005 paper â€œUncertainty, entropy, scaling and hydrological stochastics, 2, Time dependence â€¦â€ in Hydrological Sciences Journal). In the latter case statistical predictions may tend to the mean much slower than in the Markovian case and their confidence intervals would be much wider. Also, in Monte Carlo realizations, the excursions from the mean will be longer and wider.