We study the limit behaviour of upper and lower bounds on expected time\naverages in imprecise Markov chains; a generalised type of Markov chain where\nthe local dynamics, traditionally characterised by transition probabilities,\nare now represented by sets of `plausible' transition probabilities. Our first\nmain result is a necessary and sufficient condition under which these upper and\nlower bounds, called upper and lower expected time averages, will converge as\ntime progresses towards infinity to limit values that do not depend on the\nprocess' initial state. Our condition is considerably weaker than that needed\nfor ergodic behaviour; a similar notion which demands that marginal upper and\nlower expectations of functions at a single time instant converge to so-called\nlimit-or steady state-upper and lower expectations. For this reason, we refer\nto our notion as `weak ergodicity'. Our second main result shows that, as far\nas this weakly ergodic behaviour is concerned, one should not worry about which\ntype of independence assumption to adopt-epistemic irrelevance, complete\nindependence or repetition independence. The characterisation of weak\nergodicity as well as the limit values of upper and lower expected time\naverages do not depend on such a choice. Notably, this type of robustness is\nnot exhibited by the notion of ergodicity and the related inferences of limit\nupper and lower expectations. Finally, though limit upper and lower\nexpectations are often used to provide approximate information about the limit\nbehaviour of time averages, we show that such an approximation is sub-optimal\nand that it can be significantly improved by directly using upper and lower\nexpected time averages.\n