What Are the Theoretical Implications of a Time-Series Being Autoregressive?

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euroazn
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There's a bunch of literature out there about empirical/data-fitting/statistical concerns regarding autoregressive times-series models, but is there anything out there about theoretical implications of a time-series being autoregressive? For example, when does limt → ∞E(Xt) exist? Does the prediction interval as t goes to infinity have a bound?
 
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There are theoretical results - and I don't claim to know them off the top of my head!

A autoregressive model resembles a "linear recurrence relation" http://en.wikipedia.org/wiki/Recurrence_relation except it has a noise term. The solution of homegeneous linear recurrent relation is determined by finding the roots of the "auxillary equation". (Its analgous to finding the solution to a homogeneous differential equation by solving its auxillary equation.) There are theoretical results that analyze the behavior of the autoregressive model in terms of the roots of the auxilliary equation of its deterministic part. The big names in ARMA models used to be "Box Jenkins". I don't know the current theory, but "Box Jenkins" would be a good search phrase.
 
this is what I figured and of course it's easy to see that the auxiliary equation has a lot to do with expectations, but confidence intervals aren't quite so easy :(

Thanks for your help, you've been very useful!